This document provides information about circles and conic sections. It begins with an overview of circles, including definitions of key terms like radius, diameter, chord, and equations of circles given the center and radius or three points. It then covers conic sections, defining ellipses, parabolas and hyperbolas based on eccentricity. Equations of various conic sections are derived based on the location of foci, directrix, vertex and other geometric properties. Sample problems are provided to demonstrate solving problems involving different geometric configurations of circles and conic sections.
This powerpoint presentation is an introduction for the topic TRIANGLE CONGRUENCE. This topic is in Grade 8 Mathematics. I hope that you will learn something from this sides.
This document discusses different types of quadrilaterals including trapezoids, kites, parallelograms, rectangles, rhombuses, and squares. It provides definitions and properties for each shape. Key properties include:
1) A trapezoid has one pair of parallel sides, leg angles are supplementary, and the midsegment is half the sum of the bases.
2) A kite has two pairs of consecutive congruent sides, diagonals are perpendicular, and one pair of opposite angles are congruent.
3) Parallelograms have opposite sides parallel and opposite angles congruent. Rectangles and squares are types of parallelograms with right angles or all sides congruent,
The document discusses various methods for solving quadratic equations: factoring, using square roots, and completing the square. It provides examples of using each method to solve equations as well as word problems involving quadratic equations. Factoring involves writing the quadratic as a product of two binomials and setting each factor equal to 0. Using square roots involves taking the square root of both sides of the equation to isolate the variable. Completing the square rewrites the equation by adding or subtracting terms to put it in the form (x+a)2=b and then taking the square root of both sides.
This document provides information about a mathematics instructional material for Grade 9 learners in the Philippines. It was developed collaboratively by educators and reviewed by DepEd. The material covers variations, including direct, inverse, joint, and combined variations. It encourages teachers to provide feedback to DepEd to help improve the material. The material aims to help learners understand different types of variations and solve problems involving variations.
Graphs of polynomial functions are smooth and continuous, with no sharp corners or breaks. They can be drawn without lifting your pencil from the paper. A polynomial graph's behavior as x increases or decreases depends on whether its highest term has an even or odd degree. For even degrees, the graph rises or falls on both sides depending on the leading coefficient's sign. For odd degrees, the graph falls and rises or vice versa.
The document discusses functions and function notation. It provides examples of determining if relations and graphs represent functions, writing equations in function notation, evaluating functions, and using functions to model real world problems. Key points are that a relation is a function if each input is mapped to exactly one output, and a graph is a function if a vertical line intersects it at most once. Functions can be written as f(x) and evaluated by finding f(a) for some input a.
This document provides information about rational algebraic expressions. It defines a rational algebraic expression as a ratio of two polynomials where the denominator is not equal to zero. It gives examples of rational expressions like 7x/2y and 10x-5/(x+3). It explains that rational expressions are defined for all real numbers except those that would make the denominator equal to zero. The document also discusses how to simplify rational expressions by factoring the numerator and denominator and cancelling common factors. It provides examples of when cancellation is and is not allowed.
This powerpoint presentation discusses or talks about the topic or lesson Functions. It also discusses and explains the rules, steps and examples of Quadratic Functions.
The document discusses inscribed angles and intercepted arcs, explaining that the measure of an inscribed angle is equal to half the measure of its intercepted arc. It provides examples of applying this concept to find unknown angle measures. Several activities are included for students to practice identifying, measuring, and applying properties of inscribed angles and intercepted arcs.
The discriminant of a quadratic equation is represented by b^2 - 4ac. This value determines the nature of the roots. If b^2 - 4ac = 0, then the roots are real and equal. If b^2 - 4ac > 0, then the roots are real but unequal. If b^2 - 4ac < 0, then the roots are imaginary and unequal. The document provides examples of calculating the discriminant and determining the nature of roots for different quadratic equations.
Joint variation is when one quantity varies proportionally with two or more other quantities. Some key points:
- Joint variation can be represented by an equation of the form y = kxz, where y varies jointly as x and z with k as the constant of variation.
- Examples include the cost of pencils varying jointly with the number bought, and the area of a triangle varying jointly with its base and height.
- To solve a joint variation problem, the equation is set up and the constant of variation is determined using given values to find the value of the variable for a new set of quantities.
- Joint variation has many real world applications like the force on an object varying jointly with its
This learner's module discusses about the topic Variations. It also discusses the definition of Variation. It also discusses or explains the types of Variations. It also shows the examples of the Types of Variations.
This module discusses coordinate proofs and properties of circles on the coordinate plane. It introduces coordinate proofs as an analytical method of proving geometric theorems by using the coordinates of points and algebraic relationships. Examples demonstrate proving properties of triangles and quadrilaterals analytically. The standard form of the equation of a circle is derived from the distance formula as (x - h)2 + (y - k)2 = r2, where (h, k) is the center and r is the radius. Finding the center, radius, and equation of circles in various forms are illustrated.
The document discusses piecewise functions, which are functions defined by multiple rules over different parts of the domain. It provides examples of piecewise functions, including how to graph them by applying each rule over the appropriate portion of the domain. It also discusses evaluating piecewise functions at given values, writing piecewise functions based on graphs, and analyzing properties like extrema. Piecewise functions allow modeling of real-world situations using multiple equations each corresponding to a part of the input range.
The document discusses properties of hyperbolas including:
- A hyperbola is defined as the set of points where the difference between distances to two fixed points (foci) is constant.
- The standard equation of a horizontal hyperbola is (x-h)2/a2 - (y-k)2/b2 = 1 and a vertical hyperbola is (y-k)2/a2 - (x-h)2/b2 = 1.
- Examples are given to find the equation of hyperbolas given properties like the center, foci, vertices or asymptotes and to draw the graph of hyperbolas given their equation.
This document provides instructions on factoring the sum and difference of two cubes. It begins with objectives and a review of perfect cubes. It then explains that expressions like (x+4)(x^2 - 4x + 16) can be factored using the patterns for sum and difference of cubes. Two examples are worked through step-by-step to demonstrate factoring the sum and difference of two cubes using greatest common factors and rewriting the expression in terms of a and b cubes. The key steps and patterns are emphasized. It concludes with a reminder of what is important to remember from the lesson.
1. The document discusses various forms of linear equations including standard form (Ax + By = C), slope-intercept form (y = mx + b), and point-slope form (y - y1 = m(x - x1)).
2. It provides examples of transforming linear functions between these forms and discusses how to find the slope and y-intercept of a linear function from its graph.
3. The slope and direction of a linear function's graph is determined by whether its slope is positive or negative. A positive slope produces an upward rising line while a negative slope produces a downward falling line.
The document defines direct and inverse variation and provides examples of translating statements describing direct and inverse variation into mathematical equations. It also provides two examples of using the equations to solve for unknown variables given specific values of other variables. Specifically, it defines direct variation as being proportional and inverse variation as being inversely proportional. It translates statements about direct and inverse variation into equations using a constant of variation k. The examples show setting the equations equal to known values and solving for the constant k and then using k to solve for the unknown variable.
This document discusses conic sections and parabolas. It begins by listing topics that will be covered, including the geometry of parabolas and translations and reflections of parabolas. It then provides information about conic sections being the paths of objects in gravitational fields. Several slides define and show graphs of parabolas, their standard equations, and how changing parameters affects the graph. Examples are given of writing equations of parabolas given properties like the focus and directrix. A quick review section ends with practice problems finding distances and rewriting equations in standard form.
1) The document outlines the course outcomes for Calculus I with Analytic Geometry. It discusses fundamental concepts like analytic geometry, functions, limits, continuity, derivatives, and their applications.
2) The course aims to teach students to analyze and solve problems involving lines, circles, conics, transcendental functions, derivatives, tangents, normals, maxima/minima, and related rates.
3) The assessment tasks and grading criteria are also presented, including quizzes, classwork, and a final examination. Minimum averages for satisfactory performance are provided.
This document discusses concepts related to calculus including limits, continuity, and derivatives of functions. Specifically, it covers:
- Definitions and theorems related to limits, continuity, and derivatives of algebraic functions.
- Evaluating limits, determining continuity of functions, and taking derivatives of algebraic functions using basic theorems of differentiation.
- The objective is for students to be able to evaluate limits, determine continuity, and find derivatives of continuous algebraic functions in explicit or implicit form after discussing these calculus concepts.
The document provides information about conic sections, specifically circles. It defines a circle as the set of points equidistant from a fixed point, and provides the standard equation (x - h)2 + (y - k)2 = r2, where (h, k) is the circle's center and r is the radius. Several example problems are worked through, finding the equation of circles given properties like specific points or tangency to lines. The concept of a family of circles is introduced, where the circles share a common property like center location. Radical axes are defined as the line perpendicular to the line joining two circle centers.
This document discusses conic sections, which are plane curves formed by the intersection of a cone with a plane. The four types of conic sections are circles, ellipses, parabolas, and hyperbolas. It provides examples of how to determine the equation of a conic section based on given properties, such as all points being equidistant from a fixed point and line. Key aspects covered include using the distance formula and completing the square to put equations in standard form.
This document discusses ellipses. It begins by defining an ellipse as the set of all points whose distance from two fixed points, called foci, have a constant sum. It then discusses key properties of ellipses including their geometry, translations, orbits and eccentricity, and reflective property. Diagrams and equations are provided to illustrate these concepts. Examples are also included to demonstrate how to find key points and equations of ellipses.
This document provides 28 examples of graphs of different types of rational functions. The rational functions shown include those of the form y=1/x, y=-1/x, y=(x+a)/(x+b), and others with variations in the numerator and denominator polynomials. Each example graphically depicts a different rational function to illustrate their key characteristics and behaviors.
The document discusses graphing rational functions of the form f(x) = a(x)/b(x), where a(x) and b(x) are polynomial functions. It provides examples of how the graphs appear depending on whether the degree of the numerator is even or odd. A 7-step process is outlined for sketching the graphs, which involves finding intercepts, roots, vertical asymptotes, horizontal asymptotes, and the sign of the function to sketch the graph. Two examples applying the 7-step method are shown.
Vertical asymptotes to rational functionsTarun Gehlot
This document discusses how to graph rational functions by identifying key characteristics from the function expression. These include:
1) The y-intercept by setting x=0.
2) X-intercepts by setting the numerator equal to 0.
3) Vertical asymptotes by setting the denominator equal to 0.
4) Horizontal or slant asymptotes using rules based on the degrees of the numerator and denominator polynomials.
5) The graph by considering the intercepts, asymptotes, and a "sign property" that determines whether the graph is above or below the x-axis between intercepts/asymptotes. Examples are worked through step-by-step.
Rational functions have two types of asymptotes: vertical asymptotes which are found by making the denominator equal to zero, and horizontal asymptotes which come in three types based on the degrees of the numerator and denominator. To graph a rational function, you first find the vertical and horizontal asymptotes, then locate the x-intercepts, y-intercept and use a calculator to generate a table of values to graph points and draw the line. The domain is all real numbers except values making the denominator equal to zero, and the range is all real numbers.
210 graphs of factorable rational functionsmath260
The document discusses vertical asymptotes of rational functions. It provides examples of the functions y=1/x and y=1/x^2. For y=1/x, the graph does not touch the vertical asymptote at x=0, but gets infinitely close to it as x approaches 0. The graph runs upward along the right of the asymptote and downward along the left. For y=1/x^2, the graph also has a vertical asymptote at x=0 and runs upward along both sides of the asymptote. Vertical asymptotes occur where the denominator is 0.
This document discusses rational functions and their graphs. It outlines 3 cases for rational functions based on the degree of the numerator and denominator. It also discusses identifying and plotting asymptotes, including vertical, horizontal, and slant asymptotes. Methods are provided for graphing rational functions based on finding asymptotes and plotting points around the asymptotes to determine the graph's behavior. Examples are worked through demonstrating these graphing techniques.
This document provides an overview of rational functions and how to graph them. It discusses identifying vertical and horizontal asymptotes by analyzing the degrees of the numerator and denominator polynomials. It gives examples of multiplying, dividing, and graphing rational functions. Students are asked to find asymptotes, state domains and ranges, and graph rational functions. They are also given examples and problems for multiplying and dividing rational expressions.
This document discusses rational functions and their graphs. It defines a rational function as a function of the form f(x) = p(x)/q(x) where p and q are polynomials. It explains that the domain of a rational function excludes any values that would make the denominator equal to 0. It describes how to find vertical, horizontal, and oblique asymptotes of a rational function by comparing the degrees of the polynomials in the numerator and denominator. Vertical asymptotes occur where the denominator is 0, and horizontal or oblique asymptotes depend on whether the degree of the numerator is less than, equal to, or greater than the degree of the denominator. Examples are provided to illustrate these concepts.
Asymptote is a mathematical term referring to a line that a graph approaches but never meets. A vertical asymptote occurs when the function approaches positive or negative infinity as the variable increases or decreases without bound. A horizontal asymptote is a horizontal line that a function approaches but does not meet as the variable increases or decreases without bound.
1) Points of discontinuity occur when the denominator of a rational function equals zero.
2) A rational function can have at most one horizontal asymptote. If the degree of the numerator is less than the denominator, the horizontal asymptote is y=0. If the degrees are equal, the horizontal asymptote is the ratio of the leading coefficients.
3) To find the equation of a slant asymptote, divide the numerator by the denominator and keep only the terms with the highest powers of x.
This document introduces several circle theorems including:
1. Angles subtended by the same chord are equal.
2. The angle subtended by the same chord at the centre is twice that at the circumference.
3. The opposite angles of a cyclic quadrilateral add up to 180 degrees.
It also discusses properties such as an angle in a semi-circle being 90 degrees and the radius and tangent being at right angles.
The document describes a circular concrete conduit that is 1 ft thick with an inside diameter of 10 ft. The conduit rises 16 ft per 1000 horizontal ft. It has ends that are 3000 ft apart. The question asks to calculate the amount of concrete used to construct this conduit.
Rational functions are any functions that can be written as the ratio of two polynomial functions. There are two types of asymptotes for rational functions: vertical asymptotes, which occur at the zeros of the denominator and cannot be crossed, and horizontal asymptotes, which can be crossed. To find the vertical and horizontal asymptotes of a rational function, you examine the degrees of the numerator and denominator polynomials. The domain of a rational function is the set of x-values that make the function defined, while the range is the set of possible y-values produced by the function.
An asymptote is an imaginary line that a graph approaches but never touches as it goes to infinity. The document provides examples of graphs with rational functions that have vertical and horizontal asymptotes. It also discusses the square root function and how it is the inverse of the quadratic function. The square root function is only defined for non-negative inputs. Changing the parameters a and b in the square root function changes the shape and orientation of the graph.
Distinguish equations representing the circles and the conics; use the properties of a particular geometry to sketch the graph in using the rectangular or the polar coordinate system. Furthermore, to be able to write the equation and to solve application problems involving a particular geometry.
MS Report, When we talked about the conic section it involves a double-napped cone and a plane. If a plane intersects a double right circular cone, we get two-dimensional curves of different types. These curves are what we called the conic section.
The document discusses parabolas and their key properties. A parabola is defined as the set of points equidistant from a fixed point (the focus) and a fixed line (the directrix). The vertex, axis of symmetry, and focus-directrix distance determine the shape and position of the parabola. Examples are provided to demonstrate how to find the equation of a parabola given properties like the vertex and focus.
1) The document provides a refresher on analytic geometry concepts including the Cartesian plane, lines, parabolas, ellipses, and circles. It gives definitions, properties, and equations for these concepts.
2) Examples are worked through, such as finding the coordinates of points, slopes of lines, and equations of lines and circles. Practice problems and their solutions are also provided.
3) Key topics covered include the Cartesian plane, distance between points, slope and equations of lines, parallel and perpendicular lines, conic sections including parabolas, circles, and ellipses, and their defining properties and equations.
1. The document defines an ellipse and its key properties including its standard equation form. It discusses how an ellipse is a set of points where the sum of the distances to two fixed points (foci) is constant.
2. Parts of an ellipse like its vertices, covertices, axes, and directrices are defined. The standard equation of an ellipse centered at the origin is derived.
3. Examples are provided of determining the coordinates of foci, vertices, covertices, and directrices from equations. Problems involving finding equations or properties given certain conditions are also presented.
Conics are curves formed by the intersection of a plane with a double cone. A hyperbola results if the plane cuts both cones, a parabola if the plane is parallel to the edge of the cone, and an ellipse if neither of those cases apply. A circle is a special type of ellipse that occurs when the plane is perpendicular to the altitude of the cone.
Conic sections are shapes formed by the intersection of a plane and a double cone. A hyperbola occurs if the plane cuts through both cones. A parabola occurs if the plane is parallel to the edge of the cone. An ellipse occurs if the plane is not parallel or cutting through both cones. A circle is a special case of an ellipse where the plane is perpendicular to the altitude of the cone.
This document provides information about circles, including their geometric and algebraic definitions, how to write the equation of a circle in standard form, and how to graph circles. It discusses that a circle is defined geometrically as the set of all points equidistant from a fixed point, and algebraically as the set of all points with a constant distance from a fixed point. The standard form of a circle equation is given as (x-h)2 + (y-k)2 = r2, where (h,k) is the center and r is the radius. Examples are provided of writing and graphing various circle equations in this standard form.
An ellipse is defined as the set of all points where the sum of the distances from two fixed points (foci) is a constant (the length of the major axis). Key properties include:
- The vertices are the endpoints of the major axis.
- The distance from the center to each focus is the eccentricity.
- The general equation of an ellipse with center at (h,k) is (x-h)^2/a^2 + (y-k)^2/b^2 = 1.
- Examples are provided to illustrate finding the equation of an ellipse given properties like the foci, vertices, or axes.
The document discusses different topics in mathematics including conic sections, differential equations, and probability. Chapter 1 covers conic sections such as circles, parabolas, ellipses and hyperbolas. It defines a circle and discusses finding the equation of a circle given its center and radius. It also addresses finding the center and radius of a circle given its equation, finding the intersection points between two circles, and finding the equation of a circle passing through three given points.
The document discusses conic sections and ellipses. Conic sections are graphs of quadratic equations of the form Ax2 + By2 + Cx + Dy = E, where A and B are not both 0. Their graphs include circles, ellipses, parabolas and hyperbolas. Ellipses are defined as the set of all points where the sum of the distances to two fixed foci is a constant. Ellipses have a center, two axes called the semi-major and semi-minor axes, and radii along the x and y axes called the x-radius and y-radius. The standard form of an ellipse equation is presented.
Here are the steps to solve these problems:
1. The points (x1, y1) and (x2, y2) form a diameter of a circle. The point (x, y) is another point on the circle.
(a) The gradient of the diameter AB is (y2-y1)/(x2-x1).
(b) The equation of AB is y-y1 = (y2-y1)/(x2-x1)(x-x1)
(c) Since P lies on AB, substitute the point (x, y) into the equation of AB to determine the value of x.
2. A line with equation y=mx+
x2 y2
Standard Equation of hyperbola is a 2 – b2 = 1
(i) Definition hyperbola : A Hyperbola is the locus of a point in a plane which moves in the plane in such a way that the ratio of its distance from a fixed point (called focus) in the same plane to its distance from a fixed line (called directrix) is always constant which is always greater than unity.
The hyperbola whose transverse and conjugate axes are respectively the conjugate and transverse axes of a given hyperbola is called conjugate hyperbola.
Note :
(i) If e1 and e2 are the eccentricities of the
(ii) Vertices : The point A and A where the curve meets the line joining the foci S and S
hyperbola and its conjugate then
1 +
e 2 e
1 = 1
2
are called vertices of hyperbola.
(iii) Transverse and Conjugate axes : The straight line joining the vertices A and A is called transverse axes of the hyperbola. Straight line perpendicular to the transverse axes and passes through its centre called conjugate axes.
(iv) Latus Rectum : The chord of the hyperbola which passes through the focus and is perpendicular to its transverse axes is called
2b2
latus rectum. Length of latus rectum = a .
(ii) The focus of hyperbola and its1 conju2gate are concyclic.
Standard Equation and Difinitions
Ex.1 Find the equation of the hyperbola whose directrix is 2x + y = 1, focus (1,2) and
eccentricity 3 .
Sol. Let P (x,y) be any point on the hyperbola. Draw PM perpendicular from P on the directrix.
Then by definition SP = e PM
(v) Eccentricity : For the hyperbola
x2 y2
a 2 – b2
= 1,
(SP)2 = e2(PM)2
2x y 12
b2 = a2 (e2 – 1)
(x–1)2 + (y–2)2 = 3
Conjugate axes 2
5(x2 + y2 – 2x – 4y + 5} =
e = =
1
Transverse
axes
3(4x2 + y2 + 1+ 4xy – 2y – 4x)
7x2 – 2y2 + 12xy – 2x + 14y – 22 = 0
(vi) Focal distance : The distance of any point on the hyperbola from the focus is called the focal distance of the point.
Note : The difference of the focal distance of a point on the hyperbola is constant and is equal to the length
of the transverse axes. |SP – SP| = 2a (const.)
which is the required hyperbola.
Ex.2 Find the lengths of transverse axis and conjugate axis, eccentricity and the co- ordinates of foci and vertices; lengths of the latus rectum, equations of the directrices of the hyperbola 16x2 – 9y2 = –144
Sol. The equation 16x2 – 9y2 = – 144 can be
Sol. y= m1(x –a),y= m2(x + a) where m1m2 = k, given
x 2
written as 9
x2
y 2
– 16 = – 1. This is of the form
y2
In order to find the locus of their point of intersection we have to eliminate the unknown
m1 and m2. Multiplying, we get
y2 = m1m2 (x2 – a2) or y2 = k(x2–a2)
a 2 – b2 = – 1
a2 = 9, b2 = 16 a = 3, b = 4
or x – y
1 k
= a2
which represents a hyperbola.
Length of transverse axis :
The length of transverse axis = 2b = 8
Length of conjugate axis :
The length of conjugate axis = 2a = 6
5
Ex.5 T
The document discusses hyperbolas. It begins by providing an algebraic definition of a hyperbola as the set of points where the difference between the distances to two fixed points (foci) is a constant. It then provides steps for graphing a hyperbola from its standard form equation, including identifying the center, vertices, transverse/conjugate axes, asymptotes, and foci. Examples of graphing hyperbolas are shown.
The document discusses properties of parabolas, including their definition as the set of points equidistant from a focus point and directrix line. It presents the standard equation for a par
The document discusses properties and equations of circles, including the standard form of a circle equation with a given center and radius. It also discusses tangent lines to circles, providing the process and equations for finding the equation of a tangent line to a circle at a given
An ellipse is the locus of a point which moves in such a way that its distance form a fixed point is in constant ratio to its distance from a fixed line. The fixed point is called the focus and fixed line is called the directrix and the constant ratio is called the eccentricity of a ellipse denoted by (e).
In other word, we can say an ellipse is the locus of a point which moves in a plane so that the sum of it distances from fixed points is constant.
2.1 Standard Form of the equation of ellipse
Let the distance between two fixed points S and S' be 2ae and let C be the mid point of SS.
Taking CS as x- axis, C as origin.
Let P(h,k) be the moving point Let SP+ SP = 2a (fixed distance) then
(ii) Major & Minor axis : The straight line AA is called major axis and BB is called minor axis. The major and minor axis taken together are called the principal axes and its length will be given by
Length of major axis 2a Length of minor axis 2b
(iii) Centre : The point which bisect each chord of an ellipse is called centre (0,0) denoted by 'C'.
(iv) Directrix : ZM and Z M are two directrix and their equation are x= a/e and x = – a/e.
(v) Focus : S (ae, 0) and S (–ae,0) are two foci of an ellipse.
(vi) Latus Rectum : Such chord which passes through either focus and perpendicular to the major axis is called its latus rectum.
Length of Latus Rectum :
If L is (ae, 𝑙 ) then 2𝑙 is the length of
SP+S'P=
{(h ae)2 k 2} +
= 2a
Latus Rectum.
Length of Latus rectum is given by
2b2
.
h2(1– e2) + k2 = a2(1– e2)
Hence Locus of P(h, k) is given by. x2(1– e2) + y2 = a2(1– e2)
2
a
(vii) Relation between constant a, b, and e
a 2 b2
b2 = a2(1– e2) e2 =
a 2
x2
a 2 +
y
a 2 (1 e 2 ) = 1
e =
a 2
Result :
Major Axis
(a) Centre C is the point of intersection of the axes of an ellipse. Also C is the mid point of AA.
(b) Another form of standard equation of ellipse
x 2 y2
a 2 b2
1 when a < b.
Directrix Minor Axis Directrix x = -a/e x = a/e
Let us assume that a2(1– e2 )= b2
The standard equation will be given by
x2 y2
a2 b2
2.1.1 Various parameter related with standard ellipse :
In this case major axis is BB= 2b which is along y- axis and minor axis is AA= 2a along x- axis. Focus S(0,be) and S(0,–be) and directrix are y = b/e and y = –b/e.
2.2 General equation of the ellipse
The general equation of an ellipse whose focus is (h,k) and the directrix is the line ax + by + c = 0 and the eccentricity will be e. Then let P(x1,y1) be any point on the ellipse which moves such that SP = ePM
Let the equation of the ellipse x
y2
a > b
(x –h)2 + (y –k)2 =
e 2 (ax1 by1 c) 2
a 2 b2
1 1 a 2 b2
(i) Vertices of an ellipse : The point of which ellipse cut the axis x-axis at (a,0) & (– a, 0) and y- axis at (0, b) & (0, – b) is called the vertices of an ellipse.
Hence the locus of (x1,y1) will be given by (a2 + b
Here are some things you did well and could improve on:
WWW:
- You explained the key concepts around writing the equation of a circle clearly and concisely. Breaking it down step-by-step makes it easy to understand.
- Providing examples with worked solutions is very helpful for reinforcement. The visual diagrams additionally aid comprehension.
- Giving practice problems for students to try on their own, along with answers, allows for application of the material.
EBI:
- Some of the text could be formatted for easier reading (e.g. consistent formatting of equations).
- Adding brief summaries or recaps after sections of explanation may aid retention.
- Providing guidance on common errors
The document discusses circles and their equations. It defines a circle as all points in a plane that are a fixed distance from a fixed center point. This fixed distance is called the radius. The standard form of a circle equation is (x-h)2 + (y-k)2 = r2, where (h,k) are the coordinates of the center and r is the radius. It also discusses converting between the standard and general forms of a circle equation.
The document discusses the coordinate equation of a circle. It states that a circle can be defined as all points that are equidistant from the center point. The standard form of the equation of a circle is r2 = (x - h)2 + (y - k)2, where (h, k) are the coordinates of the center and r is the radius. It provides examples of writing the equation of a circle given properties like the center point and radius or a point on the circle.
This document provides two links related to calculating volumes and areas of revolution. The first link gives information on calculating the area of a surface of revolution, while the second link demonstrates how to find the volume of an object with a known cross-sectional shape that has been revolved around an axis.
This document discusses improper integrals, which represent the area under an unbounded curve or graph. There are two types of improper integrals: 1) those with an unbounded interval and 2) those with an unbounded integrand. For type 1 integrals, the interval is replaced with a variable and the limit is taken as the variable approaches infinity. For type 2 integrals, the integral is defined as the limit of the integral as it approaches the unbounded point. Several examples are provided and the reader is asked to determine if given integrals converge or diverge.
The document discusses force due to liquid pressure. It provides the formula for calculating force (F) due to liquid pressure on an area (A) at depth (h) of a liquid with density (w): F = w h A. The force increases with increases in density, depth, or area. It also provides examples of calculating total force on different objects submerged in liquids, such as a trough, dam, cube, and triangular plate.
Lesson 17 work done by a spring and pump final (1)Lawrence De Vera
The document discusses work done by springs and pumping liquids. For springs, it explains Hooke's law and provides examples of calculating work done when stretching or compressing springs using the force constant and displacement. For pumping liquids, it gives the formula for calculating work as the integral of force over distance, using examples such as pumping water out of pools, tanks, and conical or cylindrical containers.
1. This document discusses methods for calculating the length of an arc of a curve and the surface area of revolution. It provides formulas for finding arc length and surface area when curves are defined by rectangular coordinates, parametric equations, or polar coordinates.
2. Several examples are given of applying the formulas to find the arc length of curves and the surface area when graphs are revolved about axes. This includes revolving curves like y=x^3, y=x^2, and xy=2 about the x-axis and y-axis.
3. The key formulas presented are that arc length can be found using an integral of the form ∫√(dx/dy)^2 + 1 dy or
The document discusses Pappus's centroid theorem, which states that the volume of a solid generated by rotating a region R about a line l is equal to the product of the area of R and the distance traveled by R's centroid during one full rotation. The theorem can be used to calculate the volumes and surface areas of various solids of revolution. Examples are provided of applying the theorem to find volumes and surface areas of solids like cones, spheres, and others.
This document discusses finding the centroid of solids of revolution. It explains that the centroid of a solid generated by revolving a plane area about an axis will lie on that axis. To find the coordinates of the centroid, one takes the moment of an elementary disc about the coordinate axes and sums these moments by treating the discs as an integral. Two examples are worked through to demonstrate finding the coordinates of the centroid. Five exercises are then posed asking to find the centroid coordinates for solids generated by revolving given bounded regions.
This document discusses different methods for calculating the volume of solids of revolution:
- The disk method is used when the axis of revolution is part of the boundary and revolves perpendicular strips into disks. Volume is calculated by integrating the area of disks.
- The washer/ring method is used when the strips revolve perpendicular but not touching the axis, forming rings. Volume subtracts the inner radius area from the outer radius area.
- The shell method is used when strips revolve parallel to the axis, forming cylindrical shells. Volume multiplies the shell area by its thickness.
The document provides examples and homework problems for students to practice calculating volumes of solids of revolution using these different methods.
Here are the steps to find the centroid of each given plane region:
1. Region bounded by y = 10x - x^2, x-axis, x = 2, x = 5:
- Set up the integral to find the area A: ∫2^5 (10x - x^2) dx
- Evaluate the integral: A = 96
- Set up the integrals to find the x- and y-moments: ∫2^5 x(10x - x^2) dx and ∫2^5 (10x - x^2)x dx
- Evaluate the integrals: Mx = 192, My = 960
- Use the formulas for centroid: C
This document discusses calculating the area under a curve using integration. It begins by approximating the area under an irregular shape using squares and rectangles. It then introduces defining the area A as a limit of approximating rectangles as their width approaches 0. This is written as the integral from a to b of f(x) dx, where f(x) is the curve. Examples are given of setting up definite integrals to calculate the areas under curves and between two curves. Steps for determining the area of a plane figure using integration are also provided.
The document discusses various techniques for integration including integration by parts, trigonometric substitution, algebraic substitution, reciprocal substitution, and partial fraction decomposition. Integration by parts allows one to integrate products of functions. Trigonometric substitution transforms integrals into ones involving trigonometric functions that can be evaluated using basic formulas. Algebraic substitution rationalizes irrational integrals. Partial fraction decomposition expresses rational functions as sums of simpler fractions to facilitate integration.
This document provides objectives and instructions for integrating various types of functions, including:
- Rational functions using the Log Rule for Integration
- Exponential functions
- Trigonometric functions and their powers
- Functions involving inverse trigonometric functions
- Hyperbolic functions and inverse hyperbolic functions
It also gives formulas and methods for integrating specific combinations of trigonometric, exponential, and other elementary functions.
The document defines definite integrals and discusses their properties. It states that a definite integral evaluates to a single number by integrating a function over a closed interval from a lower limit to an upper limit. It gives examples of using definite integrals to find areas bounded by curves. The mean value theorem for integrals is also introduced, which states that there is a rectangle between the inscribed and circumscribed rectangles with an area equal to the region under the curve. Exercises are provided on evaluating definite integrals and applying the mean value theorem.
Lesson 7 antidifferentiation generalized power formula-simple substitutionLawrence De Vera
This document discusses antiderivatives and indefinite integration. It defines an antiderivative as a function whose derivative is the given function. The notation for antiderivatives uses an integral sign to indicate finding a function with the given differential. Several properties of indefinite integrals are described, including the power rule and that the integral of a sum is the sum of the integrals. Examples of integrating various functions using properties like substitution are provided.
This document discusses differentials and how they relate to differentiable functions. Some key points:
1. The differential of an independent variable x is defined as dx, which is equal to the increment Δx. The differential of a dependent variable y is defined as dy = f'(x) dx, where f'(x) is the derivative of the function.
2. Differentials allow approximations of changes in a function using derivatives, such as estimating errors or finding approximate roots.
3. Rules are provided for finding differentials of common functions using differentiation formulas. Examples demonstrate using differentials to estimate changes and approximate values.
This document introduces indeterminate forms and L'Hopital's rule. It defines indeterminate forms as limits that cannot be directly evaluated, such as 0/0, ∞/∞, 0×∞, etc. L'Hopital's rule states that if the limit of f(x)/g(x) is an indeterminate form, it can be evaluated by taking the limit of the derivative of the numerator over the derivative of the denominator. Several examples are provided to demonstrate applying L'Hopital's rule to evaluate limits that are indeterminate forms. The document also discusses how to handle other specific indeterminate forms like 0^0, 1^∞, and ∞-
Lesson 4 derivative of inverse hyperbolic functionsLawrence De Vera
1. The document discusses differentiation formulas for inverse hyperbolic functions including sinh-1(x), cosh-1(x), tanh-1(x), and coth-1(x).
2. Examples are provided to demonstrate taking the derivative of inverse hyperbolic functions and simplifying the results.
3. A series of exercises asks the reader to take the derivative of various functions involving inverse hyperbolic and related transcendental functions.
1. The document discusses differentiation formulas for hyperbolic functions including sinh, cosh, tanh, coth, sech, and csch. It provides examples of finding the derivatives of functions involving hyperbolic functions.
2. Hyperbolic functions are compared to trigonometric functions, noting that each pair of functions (e.g. sinh and cosh) are inverses of each other. Important hyperbolic identities are also listed.
3. Examples are given of finding the derivatives of functions involving hyperbolic functions, such as f(x) = xsinh(x). The document provides a concise review of differentiation rules for hyperbolic functions and examples of their application.
1. The differentiation formulas for the trigonometric functions sin(u), cos(u), tan(u), cot(u), sec(u), and csc(u) are presented in terms of their derivatives du/dx.
2. The derivatives are obtained by applying the basic differentiation formulas for sin(u) and cos(u) along with the quotient, product, and chain rules.
3. Formulas are also provided for deriving the derivatives of inverse trigonometric functions like arcsin, arccos, arctan, etc.
Lesson 2 derivative of inverse trigonometric functionsLawrence De Vera
This document discusses differentiation of inverse trigonometric functions. It defines inverse trigonometric functions as the inverse of trigonometric functions like sine, cosine, etc. It provides the differentiation formulas for inverse trigonometric functions by relating them to the derivatives of the original trigonometric functions using identities. Examples are also given to demonstrate finding the derivatives of various inverse trigonometric functions and simplifying the results.
The Jewish Trinity : Sabbath,Shekinah and Sanctuary 4.pdfJackieSparrow3
we may assume that God created the cosmos to be his great temple, in which he rested after his creative work. Nevertheless, his special revelatory presence did not fill the entire earth yet, since it was his intention that his human vice-regent, whom he installed in the garden sanctuary, would extend worldwide the boundaries of that sanctuary and of God’s presence. Adam, of course, disobeyed this mandate, so that humanity no longer enjoyed God’s presence in the little localized garden. Consequently, the entire earth became infected with sin and idolatry in a way it had not been previously before the fall, while yet in its still imperfect newly created state. Therefore, the various expressions about God being unable to inhabit earthly structures are best understood, at least in part, by realizing that the old order and sanctuary have been tainted with sin and must be cleansed and recreated before God’s Shekinah presence, formerly limited to heaven and the holy of holies, can dwell universally throughout creation
Lecture_Notes_Unit4_Chapter_8_9_10_RDBMS for the students affiliated by alaga...Murugan Solaiyappan
Title: Relational Database Management System Concepts(RDBMS)
Description:
Welcome to the comprehensive guide on Relational Database Management System (RDBMS) concepts, tailored for final year B.Sc. Computer Science students affiliated with Alagappa University. This document covers fundamental principles and advanced topics in RDBMS, offering a structured approach to understanding databases in the context of modern computing. PDF content is prepared from the text book Learn Oracle 8I by JOSE A RAMALHO.
Key Topics Covered:
Main Topic : DATA INTEGRITY, CREATING AND MAINTAINING A TABLE AND INDEX
Sub-Topic :
Data Integrity,Types of Integrity, Integrity Constraints, Primary Key, Foreign key, unique key, self referential integrity,
creating and maintain a table, Modifying a table, alter a table, Deleting a table
Create an Index, Alter Index, Drop Index, Function based index, obtaining information about index, Difference between ROWID and ROWNUM
Target Audience:
Final year B.Sc. Computer Science students at Alagappa University seeking a solid foundation in RDBMS principles for academic and practical applications.
About the Author:
Dr. S. Murugan is Associate Professor at Alagappa Government Arts College, Karaikudi. With 23 years of teaching experience in the field of Computer Science, Dr. S. Murugan has a passion for simplifying complex concepts in database management.
Disclaimer:
This document is intended for educational purposes only. The content presented here reflects the author’s understanding in the field of RDBMS as of 2024.
Feedback and Contact Information:
Your feedback is valuable! For any queries or suggestions, please contact muruganjit@agacollege.in
How to Install Theme in the Odoo 17 ERPCeline George
With Odoo, we can select from a wide selection of attractive themes. Many excellent ones are free to use, while some require payment. Putting an Odoo theme in the Odoo module directory on our server, downloading the theme, and then installing it is a simple process.
Beyond the Advance Presentation for By the Book 9John Rodzvilla
In June 2020, L.L. McKinney, a Black author of young adult novels, began the #publishingpaidme hashtag to create a discussion on how the publishing industry treats Black authors: “what they’re paid. What the marketing is. How the books are treated. How one Black book not reaching its parameters casts a shadow on all Black books and all Black authors, and that’s not the same for our white counterparts.” (Grady 2020) McKinney’s call resulted in an online discussion across 65,000 tweets between authors of all races and the creation of a Google spreadsheet that collected information on over 2,000 titles.
While the conversation was originally meant to discuss the ethical value of book publishing, it became an economic assessment by authors of how publishers treated authors of color and women authors without a full analysis of the data collected. This paper would present the data collected from relevant tweets and the Google database to show not only the range of advances among participating authors split out by their race, gender, sexual orientation and the genre of their work, but also the publishers’ treatment of their titles in terms of deal announcements and pre-pub attention in industry publications. The paper is based on a multi-year project of cleaning and evaluating the collected data to assess what it reveals about the habits and strategies of American publishers in acquiring and promoting titles from a diverse group of authors across the literary, non-fiction, children’s, mystery, romance, and SFF genres.
AI Risk Management: ISO/IEC 42001, the EU AI Act, and ISO/IEC 23894PECB
As artificial intelligence continues to evolve, understanding the complexities and regulations regarding AI risk management is more crucial than ever.
Amongst others, the webinar covers:
• ISO/IEC 42001 standard, which provides guidelines for establishing, implementing, maintaining, and continually improving AI management systems within organizations
• insights into the European Union's landmark legislative proposal aimed at regulating AI
• framework and methodologies prescribed by ISO/IEC 23894 for identifying, assessing, and mitigating risks associated with AI systems
Presenters:
Miriama Podskubova - Attorney at Law
Miriama is a seasoned lawyer with over a decade of experience. She specializes in commercial law, focusing on transactions, venture capital investments, IT, digital law, and cybersecurity, areas she was drawn to through her legal practice. Alongside preparing contract and project documentation, she ensures the correct interpretation and application of European legal regulations in these fields. Beyond client projects, she frequently speaks at conferences on cybersecurity, online privacy protection, and the increasingly pertinent topic of AI regulation. As a registered advocate of Slovak bar, certified data privacy professional in the European Union (CIPP/e) and a member of the international association ELA, she helps both tech-focused startups and entrepreneurs, as well as international chains, to properly set up their business operations.
Callum Wright - Founder and Lead Consultant Founder and Lead Consultant
Callum Wright is a seasoned cybersecurity, privacy and AI governance expert. With over a decade of experience, he has dedicated his career to protecting digital assets, ensuring data privacy, and establishing ethical AI governance frameworks. His diverse background includes significant roles in security architecture, AI governance, risk consulting, and privacy management across various industries, thorough testing, and successful implementation, he has consistently delivered exceptional results.
Throughout his career, he has taken on multifaceted roles, from leading technical project management teams to owning solutions that drive operational excellence. His conscientious and proactive approach is unwavering, whether he is working independently or collaboratively within a team. His ability to connect with colleagues on a personal level underscores his commitment to fostering a harmonious and productive workplace environment.
Date: June 26, 2024
Tags: ISO/IEC 42001, Artificial Intelligence, EU AI Act, ISO/IEC 23894
-------------------------------------------------------------------------------
Find out more about ISO training and certification services
Training: ISO/IEC 42001 Artificial Intelligence Management System - EN | PECB
Webinars: https://pecb.com/webinars
Article: https://pecb.com/article
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(T.L.E.) Agriculture: Essentials of GardeningMJDuyan
(𝐓𝐋𝐄 𝟏𝟎𝟎) (𝐋𝐞𝐬𝐬𝐨𝐧 𝟏.𝟎)-𝐅𝐢𝐧𝐚𝐥𝐬
Lesson Outcome:
-Students will understand the basics of gardening, including the importance of soil, water, and sunlight for plant growth. They will learn to identify and use essential gardening tools, plant seeds, and seedlings properly, and manage common garden pests using eco-friendly methods.
Beginner's Guide to Bypassing Falco Container Runtime Security in Kubernetes ...anjaliinfosec
This presentation, crafted for the Kubernetes Village at BSides Bangalore 2024, delves into the essentials of bypassing Falco, a leading container runtime security solution in Kubernetes. Tailored for beginners, it covers fundamental concepts, practical techniques, and real-world examples to help you understand and navigate Falco's security mechanisms effectively. Ideal for developers, security professionals, and tech enthusiasts eager to enhance their expertise in Kubernetes security and container runtime defenses.
Principles of Roods Approach!!!!!!!.pptxibtesaam huma
Principles of Rood’s Approach
Treatment technique used in physiotherapy for neurological patients which aids them to recover and improve quality of life
Facilitatory techniques
Inhibitory techniques
Views in Odoo - Advanced Views - Pivot View in Odoo 17Celine George
In Odoo, the pivot view is a graphical representation of data that allows users to analyze and summarize large datasets quickly. It's a powerful tool for generating insights from your business data.
The pivot view in Odoo is a valuable tool for analyzing and summarizing large datasets, helping you gain insights into your business operations.
Understanding and Interpreting Teachers’ TPACK for Teaching Multimodalities i...Neny Isharyanti
Presented as a plenary session in iTELL 2024 in Salatiga on 4 July 2024.
The plenary focuses on understanding and intepreting relevant TPACK competence for teachers to be adept in teaching multimodality in the digital age. It juxtaposes the results of research on multimodality with its contextual implementation in the teaching of English subject in the Indonesian Emancipated Curriculum.
Understanding and Interpreting Teachers’ TPACK for Teaching Multimodalities i...
Lecture co2 math 21-1
1. CO2
Math21-1
Distinguish equations representing the
circles and the conics; use the properties
of a particular geometry to sketch the
graph in using the rectangular or the polar
coordinate system. Furthermore, to be
able to write the equation and to solve
application problems involving a particular
geometry.
2. COVERAGE
Circle : Center at any point ( Include discussion on translation of axes)
3
CONICS: Properties and Application Involving the
Parabola, Ellipse and Hyperbola with
Vertex/ Center at any point with
Horizontal/Vertical/ Oblique Axis
4
Polar Curves and Parametric Curves; Sketching and Transformation to Rectangular forms of
equations
4. OBJECTIVE:
At the end of the lesson, the students should be able
to use the basic principles concerning the circle to illustrate
properly and solve diligently application problems.
5. CIRCLE
A circle is the locus of point that moves in a plane at a constant
distance from a fixed point. The fixed point is called the center,
C( h, k) and the distance from the center to any point on the
circle is called the radius, r.
The second degree equation , for some
constants C, D, and E, is the general equation of a circle.
x2
+ y2
+ Dx + Ey+ F = 0
6. ● Chord - A chord is a straight line joining two points on the
circumference. The longest chord in a circle is called a
diameter. The diameter passes through the center.
● Segment - A segment of a circle is the region enclosed by a
chord and an arc of the circle.
● Secant - A secant is a straight line cutting the circle at two
distinct points.
● Tangent - If a straight line and a circle have only one point of
contact, then that line is called a tangent. A tangent is
always perpendicular to the radius drawn to the point of
contact.
8. Let:
C (h, k) - coordinates of the center of the circle
r - radius of the circle
P (x, y) - coordinates of any point on the circle
Distance CP = radius ( r )
or r2 = (x – h)2 + (y – k)2
the center-radius form or the Standard Form of the equation
of the circle.
9. The general form of the equation of the circle is obtained from
the center-radius form (x – h)2 + (y – k)2 = r2 by expanding
the squares as follows:
(x2 – 2hx + h2) + (y2 – 2ky + k2) = r2
x2 + y2 – 2hx – 2ky + h2 + k2 - r2= 0
Comparing this form with
It can be shown that -2h = D
-2k = E
h2 + k2 - r2 = F
Thus, , and
x2
+ y2
+ Dx + Ey+ F = 0
h = -
D
2
k = -
E
2
r =
D2
+ E2
- 4F
2
10. Sample Problems
1. Find the equation of the circle satisfying the given condition:
a. center at C(3, 2) and the radius of 4 units
b. center (-1, 7) and tangent to the line 3x – 4y + 6 = 0
c. having (8, 1) and (4,-3) as ends of a diameter
d. passing through the intersection of 2x-3y+6=0 and
x+3y-6=0 with center at (3,-1).
2. Reduce the equation 4x2 + 4y2 – 4x – 8y – 31 = 0 to the
center-radius form. Draw the circle.
11. Circles Determined by Three Geometric Conditions
The standard equation and the
general equation both have three
parameters: h , k, and r for the standard equation and D, E, and F
for the general equation. Each of these equations defines a
unique circle for a given set of values of the parameters. Thus, a
unique circle results from the following conditions:
a. Given the center C ( h, k) and the radius r
b. Given 3 non-collinear points,
c. Given the equation of a tangent line, the point of
tangency, and another point on the circle
d. Given a tangent line and a pair of points on a circle
x2
+ y2
+ Dx + Ey+ F = 0
x -h( )
2
+ y-k( )
2
= r2
Pi xi, yi( )
12. Figure 1 Figure 2
Figure 3 Figure 4
Illustrations of the different conditions:
13. Sample Problems
1. Find the equation of the circle tangent to the line 4x – 3y + 12=0
at (-3, 0) and also tangent to the line 3x + 4y –16 = 0 at (4, 1).
2. Find the equation of the circle which passes through the points
(1, -2), (5, 4) and (10, 5).
3. Find the equation of the circle which passes through the points
(2, 3) and (-1, 1) , with center on the line x – 3y – 11 = 0.
4. Find the equation(s) of the circle(s) tangent to 3x-4y-4=0 at (0,-1)
and containing the point (-1,-8).
5. Find the equation of the circle tangent to the line x + y = 2 at
point (4 -2) and with center on the x-axis.
6. A triangle has its sides on the lines x + 2y – 5 = 0, 2x – y – 10 = 0
and 2x + y + 2 = 0. Find the equation of the circle inscribed in the
triangle.
14. FAMILY OF CIRCLES
A family of circles is a set of circles satisfying less than three
geometric conditions. Also, if we let x2+y2+D1x+E1y+F1=0 and
x2+y2+D2x+E2y+F2=0 be the equation of two circles and taking “k”
as the parameter, then the equation of the family of circles
passing through the intersection of two circles is
(x2+y2+D1x+E1y+F1) + k(x2+y2+D2x+E2y+F2) =0.
Except at k=-1 when the equation reduces to a linear equation
(D1-D2)x + (E1-E2)y + (F1-F2) = 0, which is called a “radical axis” of
the two given circles.
15. Sample Problems
1. Write the equation of the family of circles satisfying the given
condition:
a. With center at C (3, -4)
b. Of radius 5
c. always tangent to the Y-axis at ( 0, -6)
2. Write the equation of the family of circles C3 all members of
which pass through the intersection of the circles C1: x2+y2-
6x+2y+5=0 and C2: x2+y2-12x-2y+29=0. Find the member of the
family C3 that passes through the point (7, 0). Graph the
member of the family for which k = -1.
3. Draw the graph of the equations x2+y2-4x-6y-3=0 and x2+y2-12x-
14y+65=0. Then find the equation of the radical axis and draw
the axis.
17. OBJECTIVE:
At the end of the lesson, the students should be
able to apply the basic concepts and properties of the
conic sections in solving application problems.
18. Conic Section or a Conic is the path of a point which moves so
that its distance from a fixed point is in constant ratio to its
distance from a fixed line.
Focus is the fixed point
Directrix is the fixed line
Eccentricity ( e ) is the constant ratio given by
19. Depending on the value of the eccentricity e, the conic
sections are defined as follows:
If e = 1, the conic section is a parabola
If e < 1, the conic section is an ellipse
If e > 1, the conic section is a hyperbola
20. THE PARABOLA (e = 1)
A parabola is the set of all points in a plane equidistant
from a fixed point and a fixed line on the plane. The fixed
point is called the focus (F) and the fixed line the directrix
(D). The point midway between the focus and the directrix is
called the vertex (V). A parabola is always symmetric with
respect to its axis. The chord drawn through the focus and
perpendicular to the axis of the parabola is called the latus
rectum (LR).
22. Let: D - Directrix
F - Focus
2a - Distance from F to D
4a – the length of the Latus Rectum (LR)
(a, 0) - Coordinates of F
To derive the equation of a parabola, choose any point P on
the parabola and let it satisfy the condition
So that,
e =
PF
PD
=1
PF = PD
26. Sample Problems
1. Determine the focus, the length of the latus rectum and the equation
of the directrix for the parabola 3y2 – 8x = 0 and sketch the graph.
2. Write the equation of the parabola with vertex V at (0, 0) which
satisfies the given conditions:
a. axis on the y-axis and passes through (6, -3)
b. F(0, 4/3) and the equation of the directrix is y + 4/3 = 0
c. Directrix is x – 4 = 0
d. Focus at (0, 2)
e. Latus rectum is 6 units and the parabola opens to the left
f. Focus on the x-axis and passes through (4, 3)
3. Find the locus of the center of a circle tangent to the line y = -5 and
externally to the circle .x2
+ y2
-9 = 0
28. Consider a parabola whose axis is parallel to, but not on, a
coordinate axis. Let the vertex be at point V(h, k) and the focus
at F(h+a, k). Introduce another pair of axes by a translation of
the Origin (0, 0) to the point O’(h, k). Since the distance from
the vertex to the focus is a, the equation of the parabola on
the x’y’ plane is given by:
y’2 = 4ax’
Let x’ = x - h and y’ = y – k ( the translation formula)
Then the equation of a parabola with vertex at (h, k) and focus
at (h+a, k) on the xy plane is
(y – k)2 = 4a (x – h)
31. Standard Form General Form
(y – k)2 = 4a (x – h)
y2 + Dy + Ex + F = 0
(y – k)2 = - 4a (x – h)
(x – h)2 = 4a (y – k)
x2 + Dx + Ey + F = 0
(x – h)2 = - 4a (y – k)
32. Sample Problems
1. Draw the parabola defined by y2 + 8x – 6y + 25 = 0
2. Express x2 – 12x + 16y – 60 = 0 in standard form then draw the parabola.
3. Determine the equation of the parabola (in the standard form) satisfying
the given conditions; draw the parabola:
a. V (3, 2) and F (5, 2)
b. V (2, 3) and axis parallel to y axis and passing through (4, 5)
c. V (2, 1), Latus rectum at (-1, -5) & (-1, 7)
d. V (2, -3) and directrix is y = -7
e. with vertical axis, vertex at (-1, -2), and passes through (3, 6).
f. Axis parallel to the y-axis, passes through (1, 1), (2, 2) and (-1, 5)
g. axis parallel to the x-axis passes through (0, 4), (0, -1) and (6, 1).
33. 4. A parkway 20 meters wide is spanned by a parabolic arch 30 meters long
along the horizontal. If the parkway is centered, how high must the vertex of
the arch be in order to give a minimum clearance of 5 meters over the
parkway.
5. A parabolic suspension bridge cable is hung between two supporting towers
120 meters apart and 35 meters above the bridge deck. The lowest point of
the cable is 5 meters above the deck. Determine the lengths of the tension
members 20 meters and 40 meters from the bridge center.
6. Water spouts from a horizontal pipe 12 meters above the ground. Three
meters below the line of the pipe, the water trajectory is at a horizontal
distance of 5 meters from the water outlet. How far from the water outlet
will the stream of the water hit the ground?
7. A parabolic trough 10 meters long, 4 meters wide across the top and 3
meters deep is filled with water at a depth of 2 meters. Find the volume of
water in the trough.
34. THE ELLIPSE (e < 1)
An ellipse is the set of all points P in a plane such that
the sum of the distances of P from two fixed points F’ and F is
constant. The constant sum is equal to the length of the major
axis (2a). Each of the fixed points is called a focus (plural foci).
35. Important Terms
Eccentricity measure the degree of flatness of an ellipse. The
eccentricity of an ellipse should be less than 1.
Major axis is the segment cut by the ellipse on the line joining the
vertices of an ellipse through the foci; MA = 2a
Minor Axis is the segment cut by the ellipse on the line joining the
co-vertices through the center of the ellipse; ma = 2b
Vertices are the endpoints of the major axis.
Co-vertices are the endpoints of the minor axis
Focal chord is any chord of the ellipse through the focus.
Latus rectum ( latera recta in plural form) is the segment cut by the
ellipse through a focus and perpendicular to the major axis;
36. Properties of an Ellipse
1. The ellipse intersects the major-axis at two points called the vertices, V and V’.
2. The length of the segment VV’ is equal to 2a where a is the length of the semi-
major axis.
3. The ellipse intersects the minor axis at two points called the co-vertices, B and
B’.
4. The length of the segment BB’ is equal to 2b where b is the length of the semi-
minor axis.
5. The length of the segment FF’ is equal to 2c where c is the distance from the
center to a focus; c = ae
6. The midpoint of the segment VV’ is called the center of an ellipse denoted by
C.
7. The line segments through F1 and F2 perpendicular to the major – axis are
the latera recta and each has a length of 2b2/a.
8. The relationship of a, b and c is given by a2 = b2 + c2 where, a > b.
42. ELLIPSE WITH CENTER AT (h, k)
The equation of an ellipse on the x’y’ plane, with axes
parallel to the coordinate axes and the center at (h,k), is
given by
Using the substitutions x’ = x – h and y’ = y – k will
transform the equation to
x -h( )
2
a2
+
y -k( )
2
b2
=1
44. Sample Problems
1. Find the equation of the ellipse which satisfies the given
conditions
a. foci at (0, 4) and (0, -4) and a vertex at (0,6)
b. center (0, 0), one vertex (0, -7), one end of minor axis (5, 0)
c. foci (-5, 0), and (5, 0) length of minor axis is 8
d. foci (0, -8), and (0, 8) length of major axis is 34
e. vertices (-5, 0) and (5, 0), length of latus rectum is 8/5
f. center (2, -2), vertex (6, -2), one end of minor axis (2, 0)
g. foci (-4, 2) and (4, 2), major axis 10
h. center (5, 4), major axis 16, minor axis 10 with major axis
parallel to x-axis.
45. 2. Determine the coordinates of the foci, the ends of the major and
minor axes, and the ends of each latus rectum. Sketch the curve.
9x2 + 25y2 = 225
3. Reduce the equations to standard form. Find the coordinates of the
center, the foci, and the ends of the minor and major axes. Sketch the
graph.
a. x2 + 4y2 – 6x –16y – 32 = 0
b. 16x2 + 25y2 – 160x – 200y + 400 = 0
4. The arch of an underpass is a semi-ellipse 6m wide and 2m high. Find
the clearance at the edge of a lane if the edge is 2m from the middle.
a. b.
46. 5. The earth’s orbit is an ellipse with the sun at one focus. The
length of the major axis is 186,000,000 miles and the
eccentricity is 0.0167. Find the distances from the ends of the
major axis to the sun. These are the greatest and least distances
from the earth to the sun.
6. A hall that is 10 feet wide has a ceiling that is a semi-ellipse. The
ceiling is 10 feet high at the sides and 12 feet high in the center.
Find its equation with the x-axis horizontal and the origin at the
center of the ellipse.
47. THE HYPERBOLA (e > 1)
A hyperbola is the set of points in a plane such that the
difference of the distances of a point from two fixed points
(foci) in the plane is constant.
48. General Equation of a Hyperbola
1. Horizontal Transverse Axis : Ax2 – Cy2 + Dx + Ey + F = 0
2. Vertical Transverse Axis: Cy2 – Ax2 + Dx + Ey + F = 0
where A and C are positive real numbers
49. Important Terms and Relations
Transverse axis is a line segment joining the two vertices of the
hyperbola.
Conjugate axis is the perpendicular bisector of the transverse axis;
the line through the center joining the co-vertices.
Asymptote is a line that the hyperbola approaches to as x and y
increses without bound.
52. Then letting b2 = c2 – a2 and dividing by a2b2, we have
if foci are on the x-axis
if foci are on the y-axis
The generalized equations of hyperbola with axes parallel to the
coordinate axes and center at (h, k) are
if foci are on a line parallel to the
x-axis
if foci are on a line parallel to the
y-axis
53. Important Relations
1. a = b , a < b , a > b
2. , c = ae
3. length of Transverse Axis = 2a
4. length of Conjugate Axis = 2b
5. distance from Center to Focus = c
6. distance from Center to Vertex = a
7. distance from Center to co-vertex =b
8. length of latus rectum =
9. distance from Center to Directrix =
c2
= a2
+b2
2b2
a a2
c
55. Sample Problems
1. Find the equation of the hyperbola which satisfies the given
conditions:
a. Center (0,0), transverse axis along the x-axis, a focus at (8,0), a
vertex at (4,0).
b. Center (0, 0), conjugate axis on x-axis, one focus at , equation
of one directrix is .
c. Center (0,0), transverse axis along the x-axis, a focus at (5,0),
transverse axis = 6.
d. Center (0,0), transverse axis along y-axis, passing through the
points (5,3) and (-3,2).
e. Center (1, -2), transverse axis parallel to the y-axis, transverse axis
= 6 conjugate axis = 10.
13,0
13139y
56. f. Center (-3, 2), transverse axis parallel to the y-axis, passing through
(1,7), the asymptotes are perpendicular to each other.
g. Center (0, 6), conjugate axis along the y-axis, asymptotes are 6x –
5y + 30 = 0 and 6x + 5y – 30 = 0.
h. With transverse axis parallel to the x-axis, center at (2,-2), passing
through
i. Center at (2,-5), conjugate axis parallel to the y-axis, slopes of
asymptotes numerically one-sixteenth times the length of the latus
rectum, and distance between foci is .
j. Center (1,-1), TA // to x-axis, LR=9, DD’= .
k. Center (4,-1), TA // to y-axis, FF’=10, LR=9/2.
l. CA // to x-axis, C (3, 6), FF’= , DD’= .
m.C (-7,-2), TA // to x-axis, eccentricity= , LR=4/3.
2+3 2, 0( ) and 2+3 10, 4( )
1452
13138
56
5524
311
57. 2. Reduce each equation to its standard form. Find the coordinates
of the center, the vertices and the foci. Draw the asymptotes and
the graph of each equation.
a. 9x2 –4y2 –36x + 16y – 16 = 0
b. 49y2 – 4x2 + 48x – 98y - 291 = 0
3. Determine the equation of the hyperbola if its center is at (-4,2)
if its vertex is at (-4, 7) and the slope of an asymptote is 5/2.
58. Lesson 3 : Simplification of Equations
by Translation and Rotation of Axes
59. Translation of Axes
Consider a transformation in which the new axes, similarly
directed, are parallel to the original axes. Translation of axes is
related to performing two geometric transformations: a horizontal
shift and a vertical shift. Hence the new axes can be obtained by
shifting the old axes h units horizontally and k units vertically while
keeping their directions unchanged.
Let x and y stand for the coordinates of any point P when
referred to the old axes, and x’ and y’ the coordinates of P with
respect to the new axes then
x = x’ + h and y = y’ + k
the translation formula.
61. Sample Problems
1. Find the new coordinates of the point P(4,-2) if the origin is
moved to (-2, 3) by a translation.
2. Find the new equation of the circle x2+y2-6x+4y-3=0 after a
translation that moves the origin to the point (3,-2).
3. Translate the axes so that no first-degree term will appear in
the transformed equation.
a. x2+y2+6x-10y+12=0
b. 2x2+3y2+10x-18y+26=0
c. x2-6x-6y-15=0
62. ROTATION OF AXES
Consider a transformation in which new axes are obtained by
rotating the original axes by some positive angle . This
transformation is called rotation of axes and is used to simplify the
general second degree equation
to the form free of the product term xy and in terms of the rotated
axes given by the eqaution
q, 0 <q<90
Ax2
+Bxy+Cy2
+Dx+Ey+F =0
Ax'2
+Cy'2
+Dx'+Ey'+F =0
x' and y'
63. Identification of Conics
The general second degree equation with the product term xy
represents a conic or degenerate conic with a rotated axes based
on the discriminant value as follows; if the discriminant
vaue is:
1. less than 1, the conic is an ellipse or a circle
2. equal to 1, the conic is a parabola
3. greater than 1, the conic is a hyperbola
Ax2
+Bxy+Cy2
+Dx+Ey+F =0
B2
-4AC
64. The angle of rotation to eliminate the product term xy is
determined by . If A = C then .
The coordinates of every point P(x, y) on the graph is transformed
to the new pair P’( x’, y’) by using the rotation formula:
where
q
tan2q =
B
A-C
q = 45
x = x'cosq -y'sinq
y = x'sinq +y'cosq
cosq =
1+cos2q
2
sinq =
1-cos2q
2
66. Sample Problems
I. Identify the type of conic represented by the
equation. Then, simplify the equation to a
form free of the product term xy. Sketch the
graph of the equation.
a.
b.
c.
d.
xy = 3
x2
+ 4xy+ 4y2
-6x -5 = 0
2x2
+ xy+ y2
- 4 = 0
9x2
-24xy+16y2
-40x -30y+100 = 0
67. Lesson 4 : Polar Coordinate System
and Polar Curves
73. RELATIONS BETWEEN RECTANGULAR AND POLAR
COORDINATES
The transformation formulas that express the relationship
between rectangular coordinates and polar coordinates of a point
are as follows:
and
Also, or
;
74. Sample Problems
1. Plot the following points on a polar coordinate system:
a.
b.
c.
2. Transform the coordinate as required:
a. polar to rectangular
i. iii.
iii.
b. rectangular to polar
i. iii.
ii.
P 2,60( )
P 2,-120( )
P -4,45( )
P 1,120( )
P 3,
2p
3
æ
è
ç
ö
ø
÷
P -
1
2
,90
æ
è
ç
ö
ø
÷
P 2, 2( ) P 7,-7( )
P 2,2 3( )
75. 2
3. Write the equation as required:
A. Rectangular form of the following:
i. iv.
ii.
iii. v.
B. Polar form of the following:
i. y=2 iii.
ii. Iv. xy = 4
r = 3cosq
r = 4
r2
sin2q = 3
r =
2
1-sinq
r(2cosq +sinq) = 3
x2
+ y2
= 4
y2
= 4x
83. Standard Forms of the Polar Equations of Conics:
Let the Pole be the focus of a conic section of eccentricity e,
with directrix d units from the Focus; then the equation of the
conic is given by one of the following forms:
a. Vertical directrix , axis of symmetry
b. Horizontal directrix, axis of symmetry
r =
ed
1±ecosq
q = 0
q =
p
2
r =
ed
1±esinq
84. Sample Problems
Sketch the curve given by the following equations:
1. r = 3 5.
2. 6.
3. 7.
4. 8.
q = 2
r = 6cosq r2
= 8sin2q
r = 5cos3q r = 4+3sinq
r =
2
1-sinq
r =
8
2+ 4cosq
86. PARAMETRIC EQUATIONS
Let t be a number in an interval I. A curve is a set of ordered pairs
( x, y), where
x = f(t) and y = g(t) for all t in I .
The variable t is called the parameter and the equations x = f(t)
and y = g(t) are parametric equations of the curve.
87. Sample Problems
1. Express the equation in the rectangular form by eliminating
the parameter. Then sketch the graph
a.
b.
c.
d.
e.
x = 2t , y= - t , t Î Â
x = t , y= 2t - 1
x = 2t , y= 2t2
- t +1 , t Î Â
x = 2cost , y= 3sin t , 0 £ t <2p
x = sect , y= tan t , -
p
2
< t<
p
2
88. REFERENCES
Analytic Geometry, 6th Edition, by Douglas F. Riddle
Analytic Geometry, 7th Edition, by Gordon Fuller/Dalton Tarwater
Analytic Geometry, by Quirino and Mijares
Fundamentals of Analytic Geometry by Marquez, et al.
College Algebra and Trigonometry , 7th ed by Aufmann, Barker and
Nation