1) The document provides information on circle equations and properties including: the general form of a circle equation, finding the center and radius from an equation, and determining if a point lies inside, outside or on a circle.
2) Examples are given for writing circle equations in different forms, finding centers and radii, and finding intersection points between circles and lines.
3) The key steps for finding intersections between a line and circle are outlined using simultaneous equations and the discriminant.
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
1) This document discusses how to solve quadratic equations by graphing, including identifying the terms of a quadratic equation, finding the solutions by graphing, and graphing quadratic functions.
2) The key steps for graphing a quadratic function are to find the axis of symmetry using the standard form equation, find the vertex point, and find two other points to reflect across the axis of symmetry to complete the parabolic graph.
3) An example problem walks through graphing the quadratic equation y = x^2 - 4x by first finding the roots, vertex, and axis of symmetry, and then constructing a table to plot points and graph the parabola.
This document provides an overview of exponential and logarithmic functions. It defines one-to-one functions and inverse functions. It explains how to find the inverse of a one-to-one function and shows that the inverse of f(x) is f-1(x). Properties of exponential functions like f(x)=ax and logarithmic functions like f(x)=logax are described. The product, quotient, and power rules for logarithms are outlined along with examples. Finally, it discusses how to solve exponential and logarithmic equations using properties of these functions.
This document provides information about trigonometric functions including:
- The objectives are to convert between degrees and radians, recognize trigonometric identities, and solve trigonometric equations.
- Trigonometry has a long history dating back to ancient civilizations for measuring distances and heights. It is now widely used in fields like astronomy.
- It discusses angles, the unit circle, trigonometric ratios, special angle values, identities, conversions between degrees and radians, and solving trigonometric equations.
It is a powerpoint presentation that will help the students to enrich their knowledge about Senior High School subject of General Mathematics. It is comprised about Rational functions and its zeroes. It is also comprised of some examples and exercises to be done for the said topic.
This document provides an overview of linear functions and equations. It defines linear equations as having the standard form Ax + By = C, with examples and how to identify linear vs. nonlinear equations. Linear functions are defined as having the form f(x) = mx + b. The document discusses slope, x-intercepts, y-intercepts, and how to graph linear equations from these components. It also covers representing linear functions in slope-intercept form as y = mx + b, and point-slope form as y - y1 = m(x - x1).
This document discusses trigonometric functions and different ways of measuring angles. It introduces degrees and radians as units of angular measurement. Degrees are divided into minutes and seconds, with one degree equal to 1/360 of a full rotation. Radians are defined as the angle subtended by an arc of a circle whose length is equal to the radius. Some key conversions are provided, such as 1 radian being approximately 57.3 degrees and 2 pi radians equaling 360 degrees. Coterminal angles which share the same initial and terminal sides are also discussed.
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
1) This document discusses how to solve quadratic equations by graphing, including identifying the terms of a quadratic equation, finding the solutions by graphing, and graphing quadratic functions.
2) The key steps for graphing a quadratic function are to find the axis of symmetry using the standard form equation, find the vertex point, and find two other points to reflect across the axis of symmetry to complete the parabolic graph.
3) An example problem walks through graphing the quadratic equation y = x^2 - 4x by first finding the roots, vertex, and axis of symmetry, and then constructing a table to plot points and graph the parabola.
This document provides an overview of exponential and logarithmic functions. It defines one-to-one functions and inverse functions. It explains how to find the inverse of a one-to-one function and shows that the inverse of f(x) is f-1(x). Properties of exponential functions like f(x)=ax and logarithmic functions like f(x)=logax are described. The product, quotient, and power rules for logarithms are outlined along with examples. Finally, it discusses how to solve exponential and logarithmic equations using properties of these functions.
This document provides information about trigonometric functions including:
- The objectives are to convert between degrees and radians, recognize trigonometric identities, and solve trigonometric equations.
- Trigonometry has a long history dating back to ancient civilizations for measuring distances and heights. It is now widely used in fields like astronomy.
- It discusses angles, the unit circle, trigonometric ratios, special angle values, identities, conversions between degrees and radians, and solving trigonometric equations.
It is a powerpoint presentation that will help the students to enrich their knowledge about Senior High School subject of General Mathematics. It is comprised about Rational functions and its zeroes. It is also comprised of some examples and exercises to be done for the said topic.
This document provides an overview of linear functions and equations. It defines linear equations as having the standard form Ax + By = C, with examples and how to identify linear vs. nonlinear equations. Linear functions are defined as having the form f(x) = mx + b. The document discusses slope, x-intercepts, y-intercepts, and how to graph linear equations from these components. It also covers representing linear functions in slope-intercept form as y = mx + b, and point-slope form as y - y1 = m(x - x1).
This document discusses trigonometric functions and different ways of measuring angles. It introduces degrees and radians as units of angular measurement. Degrees are divided into minutes and seconds, with one degree equal to 1/360 of a full rotation. Radians are defined as the angle subtended by an arc of a circle whose length is equal to the radius. Some key conversions are provided, such as 1 radian being approximately 57.3 degrees and 2 pi radians equaling 360 degrees. Coterminal angles which share the same initial and terminal sides are also discussed.
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.
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 provides instruction on writing equations of circles. It begins by defining a circle as all points in the xy-plane that are a fixed distance r from a central point (h,k), known as the radius. The standard form of a circle equation is presented as (x-h)2 + (y-k)2 = r2, where (h,k) are the coordinates of the center and r is the radius. Examples are given of writing circle equations in standard form given the center and radius. Converting a circle equation from general to standard form is also demonstrated through completing the square. Homework problems are assigned from the text.
This document discusses methods for solving quadratic and cubic equations. It begins by introducing quadratic equations in standard form and methods for solving them, including factoring, completing the square, and using the quadratic formula. It then discusses properties related to the square root and applies them to solving quadratic equations. The document concludes by introducing cubic equations that are the sum or difference of cubes, and provides an example of solving one using factoring.
This document provides information about polynomial operations including:
1) Defining polynomials as algebraic expressions involving integer powers of a variable and real number coefficients.
2) Examples of adding, subtracting, and multiplying polynomials by using vertical or FOIL methods.
3) Important formulas for polynomial operations such as (a + b)(a - b) = a2 - b2 and (a + b)2 = a2 + 2ab + b2.
4) Worked examples of applying these formulas and methods to polynomials involving single and multiple variables.
This document discusses graphing quadratic functions. It defines a quadratic function as having the form y = ax^2 + bx + c, where a is not equal to 0. The graph of a quadratic function is a U-shaped parabola. It discusses finding the vertex and axis of symmetry in standard form, vertex form, and intercept form. Examples are provided for graphing quadratic functions written in these three forms.
An antiderivative of a function is a function whose derivative is the given function. The problem of antidifferentiation is interesting, complicated, and useful, especially when discussing motion.
This is the slideshow version from class.
This document discusses key concepts related to finding zeros of polynomial functions including:
1) The factor theorem, which states that a polynomial x - k is a factor of a function f(x) if and only if f(k) = 0.
2) The rational zeros theorem, which gives possible rational zeros based on the factors of the leading coefficient and constant term.
3) The fundamental theorem of algebra, which states that every polynomial of degree n has n complex zeros and examples of finding functions based on given zeros.
4) The conjugate zeros theorem, which states that if z = a + bi is a zero, then z = a - bi is also a zero for polynomials with real coefficients.
5
1) This document discusses double-angle and half-angle formulas for trigonometric functions like sine, cosine, and tangent.
2) It derives formulas that relate trig functions of double and half angles to trig functions of the original angle, such as sin(2x) = 2sin(x)cos(x) and sin(x/2) = ±√(1-cos(x))/2.
3) Examples are provided to demonstrate applying these formulas to simplify trigonometric expressions and derive new identities.
The document discusses solving quadratic equations by finding the roots or solutions of the equation. It explains that a quadratic equation is of the form ax^2 + bx + c = 0, where a ≠ 0. The roots are the values of x that make the equation equal to 0. To solve the equation, it is set equal to 0 and the square root property, that if x^2 = k then x = ±√k, is applied to find the two roots of the quadratic equation. Several examples are shown step-by-step to demonstrate solving quadratic equations to find their two roots.
Euler’s formula deals with shapes called Polyhedra.
A Polyhedron is a closed solid shape which has flat faces and straight edges.
An example of a polyhedron would be a cube, whereas a cylinder is not a polyhedron as it has curved edges.
The document is notes for a lesson on partial derivatives. It introduces partial derivatives and their motivation as slopes of curves through a point on a multi-variable function. It defines partial derivatives mathematically and gives an example. It also discusses second partial derivatives and notes that mixed partials are always equal due to Clairaut's Theorem when the function is continuous. Finally, it provides an example of calculating second partial derivatives.
Complementary angles add up to 90 degrees, while supplementary angles add up to 180 degrees. The document provides examples of finding complementary angles by subtracting an acute angle from 90 degrees, and finding supplementary angles by subtracting an acute angle from 180 degrees. It concludes by thanking the audience.
The document discusses the relationship between the roots, solutions, zeros, x-intercepts, and factors of polynomial functions. It explains that the roots of a polynomial are the solutions to the polynomial equation when set equal to zero. The roots are also the x-intercepts of the graph of the polynomial function. Finding the roots involves factoring the polynomial and setting each factor equal to zero, or using theorems like the Fundamental Theorem of Algebra.
This document discusses circles in the coordinate plane. It defines the standard form of a circle equation as (x-h)2 + (y-k)2 = r2, where (h,k) are the coordinates of the center and r is the radius. Examples are given of identifying the center and radius from given equations and writing equations from given center and radius or a point on the circle. The steps for graphing a circle are outlined as plotting the center and drawing points at a distance of the radius from the center to form the circle.
The document discusses solving quadratic equations. It provides examples of solving quadratic equations by factoring, completing the square, and using the quadratic formula. Various techniques are demonstrated including finding the solutions sets for quadratic equations.
AS LEVEL Function (CIE) EXPLAINED WITH EXAMPLE AND DIAGRAMSRACSOelimu
1. A function is a relationship between inputs and outputs where each input corresponds to exactly one output. The domain of a function is the set of possible inputs, and the range is the set of possible outputs.
2. For a function to be one-to-one, each output must correspond to only one input. This can be tested using the horizontal line test - drawing horizontal lines on the graph. Restricting the domain can make non-one-to-one functions one-to-one.
3. The inverse of a function undoes the input-output relationship by switching the domain and range. Only one-to-one functions have inverses. The graph of an inverse function passes the vertical
The document discusses antiderivatives and integration. It defines an antiderivative as a function whose derivative is the original function. The integral of a function is defined as the set of its antiderivatives. Basic integration rules are provided, such as integrating term-by-term and pulling out constants. Formulas for integrating common functions like exponentials, trigonometric functions, and logarithms are listed. An example problem demonstrates finding the antiderivative of a multi-term function by applying the basic integration rules.
Circles are geometric shapes defined as the set of all points in a plane that are equidistant from a fixed point called the center. The distance from the center to any point on the circle is called the radius. Circle equations can be written in standard form (x-h)2 + (y-k)2 = r2, where (h, k) are the coordinates of the center and r is the radius. Problems involve finding the equation of a circle given its center and radius, graphing circles, rewriting equations in standard form, determining if an equation represents a circle, point or null set, finding equations of circles tangent to lines or passing through points, and finding equations of circles inscribed in or
This document provides information about circles and their relationships to other geometric concepts:
1. It defines key concepts such as the distance formula, points on a circle, the equation of a circle with the center at the origin or other point (x,y), and the general equation of a circle.
2. It discusses the intersection of circles and lines, including finding the points of intersection using substitution and solving quadratics. The discriminant is used to determine the number of intersection points.
3. Tangents are defined as lines that intersect a circle at only one point and formulas are given for finding the equation of a tangent line to a circle.
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.
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 provides instruction on writing equations of circles. It begins by defining a circle as all points in the xy-plane that are a fixed distance r from a central point (h,k), known as the radius. The standard form of a circle equation is presented as (x-h)2 + (y-k)2 = r2, where (h,k) are the coordinates of the center and r is the radius. Examples are given of writing circle equations in standard form given the center and radius. Converting a circle equation from general to standard form is also demonstrated through completing the square. Homework problems are assigned from the text.
This document discusses methods for solving quadratic and cubic equations. It begins by introducing quadratic equations in standard form and methods for solving them, including factoring, completing the square, and using the quadratic formula. It then discusses properties related to the square root and applies them to solving quadratic equations. The document concludes by introducing cubic equations that are the sum or difference of cubes, and provides an example of solving one using factoring.
This document provides information about polynomial operations including:
1) Defining polynomials as algebraic expressions involving integer powers of a variable and real number coefficients.
2) Examples of adding, subtracting, and multiplying polynomials by using vertical or FOIL methods.
3) Important formulas for polynomial operations such as (a + b)(a - b) = a2 - b2 and (a + b)2 = a2 + 2ab + b2.
4) Worked examples of applying these formulas and methods to polynomials involving single and multiple variables.
This document discusses graphing quadratic functions. It defines a quadratic function as having the form y = ax^2 + bx + c, where a is not equal to 0. The graph of a quadratic function is a U-shaped parabola. It discusses finding the vertex and axis of symmetry in standard form, vertex form, and intercept form. Examples are provided for graphing quadratic functions written in these three forms.
An antiderivative of a function is a function whose derivative is the given function. The problem of antidifferentiation is interesting, complicated, and useful, especially when discussing motion.
This is the slideshow version from class.
This document discusses key concepts related to finding zeros of polynomial functions including:
1) The factor theorem, which states that a polynomial x - k is a factor of a function f(x) if and only if f(k) = 0.
2) The rational zeros theorem, which gives possible rational zeros based on the factors of the leading coefficient and constant term.
3) The fundamental theorem of algebra, which states that every polynomial of degree n has n complex zeros and examples of finding functions based on given zeros.
4) The conjugate zeros theorem, which states that if z = a + bi is a zero, then z = a - bi is also a zero for polynomials with real coefficients.
5
1) This document discusses double-angle and half-angle formulas for trigonometric functions like sine, cosine, and tangent.
2) It derives formulas that relate trig functions of double and half angles to trig functions of the original angle, such as sin(2x) = 2sin(x)cos(x) and sin(x/2) = ±√(1-cos(x))/2.
3) Examples are provided to demonstrate applying these formulas to simplify trigonometric expressions and derive new identities.
The document discusses solving quadratic equations by finding the roots or solutions of the equation. It explains that a quadratic equation is of the form ax^2 + bx + c = 0, where a ≠ 0. The roots are the values of x that make the equation equal to 0. To solve the equation, it is set equal to 0 and the square root property, that if x^2 = k then x = ±√k, is applied to find the two roots of the quadratic equation. Several examples are shown step-by-step to demonstrate solving quadratic equations to find their two roots.
Euler’s formula deals with shapes called Polyhedra.
A Polyhedron is a closed solid shape which has flat faces and straight edges.
An example of a polyhedron would be a cube, whereas a cylinder is not a polyhedron as it has curved edges.
The document is notes for a lesson on partial derivatives. It introduces partial derivatives and their motivation as slopes of curves through a point on a multi-variable function. It defines partial derivatives mathematically and gives an example. It also discusses second partial derivatives and notes that mixed partials are always equal due to Clairaut's Theorem when the function is continuous. Finally, it provides an example of calculating second partial derivatives.
Complementary angles add up to 90 degrees, while supplementary angles add up to 180 degrees. The document provides examples of finding complementary angles by subtracting an acute angle from 90 degrees, and finding supplementary angles by subtracting an acute angle from 180 degrees. It concludes by thanking the audience.
The document discusses the relationship between the roots, solutions, zeros, x-intercepts, and factors of polynomial functions. It explains that the roots of a polynomial are the solutions to the polynomial equation when set equal to zero. The roots are also the x-intercepts of the graph of the polynomial function. Finding the roots involves factoring the polynomial and setting each factor equal to zero, or using theorems like the Fundamental Theorem of Algebra.
This document discusses circles in the coordinate plane. It defines the standard form of a circle equation as (x-h)2 + (y-k)2 = r2, where (h,k) are the coordinates of the center and r is the radius. Examples are given of identifying the center and radius from given equations and writing equations from given center and radius or a point on the circle. The steps for graphing a circle are outlined as plotting the center and drawing points at a distance of the radius from the center to form the circle.
The document discusses solving quadratic equations. It provides examples of solving quadratic equations by factoring, completing the square, and using the quadratic formula. Various techniques are demonstrated including finding the solutions sets for quadratic equations.
AS LEVEL Function (CIE) EXPLAINED WITH EXAMPLE AND DIAGRAMSRACSOelimu
1. A function is a relationship between inputs and outputs where each input corresponds to exactly one output. The domain of a function is the set of possible inputs, and the range is the set of possible outputs.
2. For a function to be one-to-one, each output must correspond to only one input. This can be tested using the horizontal line test - drawing horizontal lines on the graph. Restricting the domain can make non-one-to-one functions one-to-one.
3. The inverse of a function undoes the input-output relationship by switching the domain and range. Only one-to-one functions have inverses. The graph of an inverse function passes the vertical
The document discusses antiderivatives and integration. It defines an antiderivative as a function whose derivative is the original function. The integral of a function is defined as the set of its antiderivatives. Basic integration rules are provided, such as integrating term-by-term and pulling out constants. Formulas for integrating common functions like exponentials, trigonometric functions, and logarithms are listed. An example problem demonstrates finding the antiderivative of a multi-term function by applying the basic integration rules.
Circles are geometric shapes defined as the set of all points in a plane that are equidistant from a fixed point called the center. The distance from the center to any point on the circle is called the radius. Circle equations can be written in standard form (x-h)2 + (y-k)2 = r2, where (h, k) are the coordinates of the center and r is the radius. Problems involve finding the equation of a circle given its center and radius, graphing circles, rewriting equations in standard form, determining if an equation represents a circle, point or null set, finding equations of circles tangent to lines or passing through points, and finding equations of circles inscribed in or
This document provides information about circles and their relationships to other geometric concepts:
1. It defines key concepts such as the distance formula, points on a circle, the equation of a circle with the center at the origin or other point (x,y), and the general equation of a circle.
2. It discusses the intersection of circles and lines, including finding the points of intersection using substitution and solving quadratics. The discriminant is used to determine the number of intersection points.
3. Tangents are defined as lines that intersect a circle at only one point and formulas are given for finding the equation of a tangent line to a circle.
This document provides a study package on circles for a mathematics class. It begins with an index listing the topics covered, which include theory, revision, exercises, assertion and reason questions, and past examination questions. It then covers circle theory, equations of circles in various forms including parametric and Cartesian, intercepts made by circles on axes, the position of points with respect to circles, lines and circles, and tangents to circles. Examples are provided to illustrate each concept. The document is intended to be a comprehensive resource for students to learn about circles.
The document discusses equations of circles in various forms. It provides the general equation of a circle, as well as equations for circles given specific properties like center point and radius, diameter endpoints, tangency to an axis, and passing through a given point. Examples are worked through to find the equation of a circle matching given conditions or to obtain properties of a circle from its equation. Circles can be represented using forms based on the center and radius, diameter endpoints, or general equation.
Conic sections are shapes formed by the intersection of a plane and a double-napped cone. The four types of conic sections are circles, parabolas, ellipses, and hyperbolas. A circle is defined as the set of all points equidistant from a fixed center point in a plane. The standard equation of a circle is (x-h)2 + (y-k)2 = r2, where (h,k) represents the center and r is the radius. This document provides examples of writing and graphing circle equations in standard and general form.
This document provides examples of finding the center and radius of circles from their equations. It shows that if a circle equation is given as x^2 + y^2 + 2gx + 2fy + c = 0, then the center is (-g, -f) and the radius is the square root of g^2 + f^2 - c. It works through multiple examples, equating coefficients to find g, f, and c, and then uses this formula to determine the center and radius.
- The general equation of a circle is x2 + y2 + 2gx + 2fy + c = 0.
- Comparing this equation to the standard form (x - h)2 + (y - k)2 = r2 allows us to determine the circle's center (h, k) and radius r from any circle equation.
- For the example circle x2 + y2 + 4x - 6y + 4 = 0, the center is (-2, 3) and the radius is 3 units.
This document contains 5 math problems involving factorizing expressions, solving equations, evaluating expressions for given values, expanding expressions, and finding the highest common factor. It also provides context on working with straight line graphs, including finding the gradient and y-intercept of a line from its equation, finding the gradient between two points, finding the midpoint and a point that divides a line segment in a given ratio, and finding the x- and y-intercepts of a line.
Circle, Definition, Equation of circle whose center and radius is known, General equation of a circle, Equation of circle passing through three given points, Equation of circle whose diameters is line joining two points (x1, y1) & (x2,y2), Tangent and Normal to a given circle at given point.
2.7 more parabolas a& hyperbolas (optional) tmath260
The document provides examples of how to graph and identify properties of hyperbolas and parabolas. It includes:
1) An example of a hyperbola with center (3, -1), x-radius of 4, y-radius of 2, and vertices of (7, -1) and (-1, -1).
2) Steps for completing the square to put a quadratic equation into standard form for a hyperbola, including an example starting with 4y^2 - 9x^2 - 18x - 16y = 29.
3) An example of graphing a parabola given by the equation x = -y^2 + 2y + 15, identifying
35182797 additional-mathematics-form-4-and-5-notesWendy Pindah
1) The function f(x) = 2x^2 + 8x + 6 can be written as f(x) = 2(x+2)^2 - 2. The maximum point is (-2, -2) and the equation of the tangent at this point is y = -2.
2) The function f(x) = -(x-4)^2 + h has a maximum point at (k, 9) so k = 4 and h = 9.
3) The function y = (x+m)^2 + n has an axis of symmetry at x = -m. Given the axis is x = 1, m = -1 and the minimum point is (1
The document presents several problems involving finding equations of circles given properties such as the center and radius, points the circle passes through, tangency to lines, etc. There are over 20 subproblems across 4 sections - A focuses on finding center and radius of circles given equations, B focuses on finding equations of circles given properties like center and radius or points, C involves circles tangent to given lines, and D analyzes if equations represent circles, points or empty sets and finds properties if they are circles.
The document provides information on exam format and topics that need to be studied for Form 4 and Form 5 exams.
It recommends setting targets and being familiar with exam format. The main topics covered are functions, quadratic equations, trigonometry, calculus, vectors, statistics, and index numbers. Exercise and practice are strongly emphasized. Sample exam papers and questions are provided to illustrate exam structure and level of difficulty.
Write the equation of a circle given the center and radius
Identify the center and radius of a circle in both center-radius and general form
Write the equation of a circle given the center and a point on the circle
This document discusses distance, circles, and quadratic equations in three parts:
1) It derives the formula for finding the distance between two points in a plane as the square root of the sum of the squares of the differences of their x- and y-coordinates.
2) It derives the midpoint formula for finding the midpoint between two points as the average of their x-coordinates and the average of their y-coordinates.
3) It discusses the standard equation of a circle, gives methods for finding the center and radius from different forms of the circle equation, and notes degenerate cases where the equation does not represent a circle.
The document discusses the standard equation of a circle. It explains that if the circle is centered at the origin (0,0), the equation is x2 + y2 = r2, where r is the radius. If the circle is not centered at the origin, the equation is (x - h)2 + (y - k)2 = r2, where (h, k) are the coordinates of the center and r is the radius. The document provides examples of writing the equation of a circle given the center and radius. It also demonstrates how to determine if a point lies inside, outside, or on the circle given the equation.
1. The document presents problems involving number theory, algebra, geometry, and probability. For number theory, it provides exercises and solutions involving sums of powers and nearest integers. For algebra, it solves systems of equations and determines values based on relationships between variables. For geometry, it calculates areas and volumes. For probability, it determines probabilities of events occurring based on arrangements and selections from sets.
1. The document presents problems involving number theory, algebra, geometry, and probability. For number theory, it provides exercises and solutions involving sums of powers and finding the nearest integer of a difference. For algebra, it solves systems of equations and determines values based on given equations. For geometry, it calculates areas and volumes. For probability, it finds probabilities of arrangements and outcomes of dice rolls and ball draws.
The document provides information about quadratic functions including:
- How to find the roots, intercept, turning point, and whether the turning point is a maximum or minimum value by analyzing the quadratic equation.
- Examples of using factorizing, completing the square, and setting the equation equal to 0 to find the roots.
- How to sketch the graph of a quadratic function using its equation by plotting the roots, intercept, turning point, and labeling whether the turning point is a max or min.
The document provides information about trigonometric identities of the form sin(A+B) and double angle formulae. It includes:
1) Trigonometric identity formulas for sin(A+B), sin(A-B), cos(A+B), and cos(A-B).
2) Examples of using the identity formulas to simplify trigonometric expressions and prove identities.
3) Double angle formulas for sin(2A), cos(2A) and their uses in finding exact trig values and solving trig equations.
This document provides information about integration in higher mathematics. It begins with an overview of integration as the opposite of differentiation. It then discusses using antidifferentiation to find integrals by reversing the power rule for differentiation. Several examples are provided to illustrate integrating polynomials. The document also discusses using integrals to find the area under a curve or between two curves. It provides examples of calculating areas bounded by graphs and the x-axis. Finally, it presents some exam-style integration questions for practice.
The document provides information about polynomials at Higher level, including:
- Definitions of polynomials and examples of polynomial expressions
- Evaluating polynomials using substitution and nested/synthetic methods
- The factor theorem and using it to factorize polynomials
- Finding missing coefficients in polynomials
- Finding the polynomial expression given its zeros
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তাই একজন নাগরিক হিসাবে এই তথ্য গুলো আপনার জানা প্রয়োজন ...।
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1. Higher Unit 2
www.mathsrevision.com
Higher
Outcome 4
The Graphical Form of the Circle Equation
Inside , Outside or On the Circle
Intersection Form of the Circle Equation
Finding distances involving circles and lines
Find intersection points between a Line & Circle
Tangency (& Discriminant) to the Circle
Equation of Tangent to the Circle
Mind Map of Circle Chapter
www.mathsrevision.com
Exam Type Questions
3. Equation of a Circle
Centre at the Origin
By Pythagoras Theorem
y-axis
c
OP has length r
r is the radius of the circle
(x2 + y2 ) = r 2
b
a
a2+b2=c2
P(x,y)
y
r
O
Feb 2, 2014
x
www.mathsrevision.com
x-axis
3
5. General Equation of a Circle
y-axis
y
CP has length r
r is the radius of the circle
P(x,y)
r
y-b
C(a,b)
b
( x − a ) 2 + ( y − b) 2 = r 2
x-a
O
a
c
b
a
a2+b2=c2
with centre (a,b)
By Pythagoras Theorem
Centre C(a,b)
x
x-axis
To find the equation of a circle you need to know
Centre C (a,b) and radius r
OR
Centre C (a,b) and point on the circumference of the circle
Feb 2, 2014
www.mathsrevision.com
5
6. The Circle
Higher
Examples
Outcome 4
www.mathsrevision.com
(x-2)2 + (y-5)2 = 49 centre (2,5) radius = 7
(x+5)2 + (y-1)2 = 13 centre (-5,1) radius = √13
centre (3,0) radius = √20 = √4 X
√5
= 2√5
Centre (2,-3) & radius = 10
(x-3)2 + y2 = 20
Equation is (x-2)2 + (y+3)2 = 100
NAB
Centre (0,6) & radius = 2√3
r2 = 2√3 X 2√3
= 4√9
= 12
Equation is x2 + (y-6)2 = 12
7. The Circle
www.mathsrevision.com
Higher
Outcome 4
Example
C
Q
P Find the equation of the circle that has PQ
as diameter where P is(5,2) and Q is(-1,-6).
C is ((5+(-1))/2,(2+(-6))/2)
= (2,-2) =
CP2 = (5-2)2 + (2+2)2
= 9 + 16 = 25 = r2
Using
(x-a)2 + (y-b)2 = r2
Equation is (x-2)2 + (y+2)2 = 25
(a,b)
8. The Circle
www.mathsrevision.com
Higher
Example
Outcome 4
Two circles are concentric. (ie have same centre)
The larger has equation (x+3)2 + (y-5)2 = 12
The radius of the smaller is half that of the larger.
Find its equation.
Using
(x-a)2 + (y-b)2 = r2
Centres are at (-3, 5)
Larger radius = √12 = √4 X √3
Smaller radius = √3 so
= 2 √3
r2 = 3
Required equation is (x+3)2 + (y-5)2 = 3
9. Inside / Outside or On Circumference
Outcome 4
www.mathsrevision.com
Higher
When a circle has equation
(x-a)2 + (y-b)2 = r2
If (x,y) lies
on the circumference then
(x-a)2 + (y-b)2 = r2
If (x,y) lies
inside the circumference then
(x-a)2 + (y-b)2 < r2
If (x,y) lies
outside the circumference then (x-a)2 + (y-b)2 > r2
Example
Taking the circle
(x+1)2 + (y-4)2 = 100
Determine where the following points lie;
K(-7,12) , L(10,5) , M(4,9)
10. Inside / Outside or On Circumference
Outcome 4
www.mathsrevision.com
Higher
At K(-7,12)
(x+1)2 + (y-4)2 = (-7+1)2 + (12-4)2 = (-6)2 + 82 = 36 + 64
So point K is on the circumference.
= 100
At L(10,5)
(x+1)2 + (y-4)2 = (10+1)2 + (5-4)2 = 112 + 12 = 121 + 1 = 122
> 100
So point L is outside the circumference.
At M(4,9)
(x+1)2 + (y-4)2 = (4+1)2 + (9-4)2 = 52 + 52 = 25 + 25 = 50
So point M is inside the circumference.
< 100
11. Intersection Form of the Circle Equation
2
2
2
1. ( x − a ) + ( y − b) = r
Centre C(a,b) Radius r
( x − a )( x − a ) + ( y − b)( y − b) = r 2
( x − 2 xa + a ) + ( y − 2 yb + b ) = r
2
2
2
2
2
x 2 + y 2 − 2 xa − 2 yb + a 2 + b 2 = r 2
x 2 + y 2 − 2ax − 2by + a 2 + b 2 − r 2 = 0
Let
g = - a,
f = -b,
c = (-g)2 + ( −f) 2 − r 2
c = g 2 + f2 − r 2
r 2 = g2 + f2 − c
r = g2 + f2 − c
c = a 2 + b2 − r2
2. x 2 + y 2 + 2 gx + 2 fy + c = 0
Feb 2, 2014
c = a 2 + b2 − r2
Centre C(-g,-f) Radius r = g 2 + f 2 − c
www.mathsrevision.com
11
12. Equation x2 + y2 + 2gx + 2fy + c = 0
www.mathsrevision.com
Higher
Example
Outcome 4
Write the equation (x-5)2 + (y+3)2 = 49 without brackets.
(x-5)2 + (y+3)2 = 49
(x-5)(x+5) + (y+3)(y+3) = 49
x2 - 10x + 25 + y2 + 6y + 9 – 49 = 0
x2 + y2 - 10x + 6y -15 = 0
This takes the form given above where
2g = -10 , 2f = 6 and c = -15
13. Equation x2 + y2 + 2gx + 2fy + c = 0
www.mathsrevision.com
Higher
Example
Outcome 4
Show that the equation
x2 + y2 - 6x + 2y - 71 = 0
represents a circle and find the centre and radius.
x2 + y2 - 6x + 2y - 71 = 0
x2 - 6x + y2 + 2y = 71
(x2 - 6x + 9) + (y2 + 2y + 1) = 71 + 9 + 1
(x - 3)2 + (y + 1)2 = 81
This is now in the form (x-a)2 + (y-b)2 = r2
So represents a circle with centre (3,-1) and radius = 9
14. Equation x2 + y2 + 2gx + 2fy + c = 0
www.mathsrevision.com
Higher
Outcome 4
Example
We now have 2 ways on finding the centre and radius of a circle
depending on the form we have.
x2 + y2 - 10x + 6y - 15 = 0
2g = -10
g = -5
centre = (-g,-f)
= (5,-3)
2f = 6
f=3
c = -15
radius = √(g2 + f2 – c)
= √(25 + 9 – (-15))
= √49
= 7
16. Equation x2 + y2 + 2gx + 2fy + c = 0
Higher
Outcome 4
Example
www.mathsrevision.com
Find the centre & radius of
x2 + y2 - 10x + 4y - 5 = 0
x2 + y2 - 10x + 4y - 5 = 0
2g = -10
g = -5
centre = (-g,-f) = (5,-2)
2f = 4
f=2
NAB
c = -5
radius = √(g2 + f2 – c)
= √(25 + 4 – (-5))
= √34
17. Equation x2 + y2 + 2gx + 2fy + c = 0
www.mathsrevision.com
Higher
Example
Outcome 4
The circle x2 + y2 - 10x - 8y + 7 = 0
cuts the y- axis at A & B. Find the length of AB.
At A & B x = 0
Y
A
B
so the equation becomes
y2 - 8y + 7 = 0
(y – 1)(y – 7) = 0
y = 1 or y = 7
A is (0,7) & B is (0,1)
So AB = 6 units
18. Application of Circle Theory
Outcome 4
www.mathsrevision.com
Higher
Frosty the Snowman’s lower body section can be represented
by the equation
x2 + y2 – 6x + 2y – 26 = 0
His middle section is the same size as the lower but his
head is only 1/3 the size of the other two sections. Find
the equation of his head !
x2 + y2 – 6x + 2y – 26 = 0
2g = -6
g = -3
2f = 2
f=1
c = -26
centre = (-g,-f) = (3,-1)
radius = √(g2 + f2 – c)
= √(9 + 1 + 26)
= √36
= 6
19. Working with Distances
Outcome 4
Higher
www.mathsrevision.com
(3,19)
2
6
radius of head = 1/3 of 6 = 2
Using
(3,11)
6
6
(3,-1)
Equation is
(x-a)2 + (y-b)2 = r2
(x-3)2 + (y-19)2 = 4
20. Working with Distances
www.mathsrevision.com
Higher
Outcome 4
Example
By considering centres and radii prove that the following two
circles touch each other.
Circle 1
Circle 2
Circle 1
x2 + y2 + 4x - 2y - 5 = 0
x2 + y2 - 20x + 6y + 19 = 0
2g = 4 so g = 2
2f = -2 so f = -1
c = -5
centre = (-g, -f)
= (-2,1)
radius = √(g2 + f2 – c)
= √(4 + 1 + 5)
= √10
Circle 2
2g = -20 so g = -10
2f = 6 so f = 3
c = 19
centre = (-g, -f)
= (10,-3)
radius = √(g2 + f2 – c)
= √(100 + 9 – 19)
= √90
= √9 X √10 = 3√10
21. Working with Distances
Outcome 4
Higher
www.mathsrevision.com
If d is the distance between the centres then
d2 = (x2-x1)2 + (y2-y1)2 = (10+2)2 + (-3-1)2 = 144 + 16
= 160
d = √160
= √16 X √10 = 4√10
r2
r1
radius1 + radius2
= √10 + 3√10
= 4√10
= distance between centres
It now follows
that the circles touch
!
22. Intersection of Lines & Circles
Outcome 4
www.mathsrevision.com
Higher
There are 3 possible scenarios
2 points of
contact
discriminant
(b2- 4ac > 0)
1 point of contact
line is a tangent
discriminant
(b2- 4ac = 0)
0 points of contact
discriminant
(b2- 4ac < 0)
To determine where the line and circle meet we use simultaneous
equations and the discriminant tells us how many solutions we have.
23. Intersection of Lines & Circles
www.mathsrevision.com
Higher
Outcome 4
Example
Find where the line y = 2x + 1 meets the circle
(x – 4)2 + (y + 1)2 = 20 and comment on the answer
Replace y by 2x + 1 in the circle equation
(x – 4)2 + (y + 1)2 = 20
becomes
(x – 4)2 + (2x + 1 + 1)2 = 20
(x – 4)2 + (2x + 2)2 = 20
x 2 – 8x + 16 + 4x 2 + 8x + 4 = 20
5x 2 = 0
x2 =0
x = 0 one solution tangent point
Using
y = 2x + 1, if x = 0 then y = 1
Point of contact is (0,1)
24. Intersection of Lines & Circles
www.mathsrevision.com
Higher
Outcome 4
Example
Find where the line y = 2x + 6 meets the circle
x2 + y2 + 10x – 2y + 1 = 0
Replace y by 2x + 6 in the circle equation x2 + y2 + 10x – 2y + 1 = 0
becomes
x2 + (2x + 6)2 + 10x – 2(2x + 6) + 1 = 0
x 2 + 4x2 + 24x + 36 + 10x – 4x - 12 + 1 = 0
5x2 + 30x + 25 = 0 ( ÷5 )
x 2 + 6x + 5 = 0
(x + 5)(x + 1) = 0
x = -5 or x = -1
Using y = 2x + 6
if x = -5 then y = -4
if x = -1 then y = 4
Points of contact
are
(-5,-4) and (-1,4).
25. Tangency
www.mathsrevision.com
Higher
Example
Outcome 4
Prove that the line 2x + y = 19 is a tangent to the circle
x2 + y2 - 6x + 4y - 32 = 0 , and also find the point of contact.
2x + y = 19 so y = 19 – 2x
Replace y by (19 – 2x) in the circle equation.
NAB
x2 + y2 - 6x + 4y - 32 = 0
x2 + (19 – 2x)2 - 6x + 4(19 – 2x) - 32 = 0
x2 + 361 – 76x + 4x2 - 6x + 76 – 8x - 32 = 0
5x2 – 90x + 405 = 0 ( ÷5)
Using
x2 – 18x + 81 = 0
If x = 9 then y = 1
(x – 9)(x – 9) = 0
Point of contact is (9,1)
x = 9 only one solution hence tangent
y = 19 – 2x
26. Using Discriminants
Outcome 4
www.mathsrevision.com
Higher
At the line x2 – 18x + 81 = 0 we can also show there is only
one solution by showing that the discriminant is zero.
For x2 – 18x + 81 = 0 ,
So
a =1, b = -18 and c = 9
b2 – 4ac = (-18)2 – 4 X 1 X 81 = 364 - 364 = 0
Since disc = 0 then equation has only one root so there
is only one point of contact so line is a tangent.
The next example uses discriminants in a slightly
different way.
27. Using Discriminants
www.mathsrevision.com
Higher
Outcome 4
Example
Find the equations of the tangents to the circle x 2 + y2 – 4y – 6 = 0
from the point (0,-8).
x2 + y2 – 4y – 6 = 0
2g = 0 so g = 0
2f = -4 so f = -2
Centre is (0,2)
Y
(0,2)
Each tangent takes the form y = mx -8
Replace y by (mx – 8) in the circle equation
to find where they meet. This gives us …
x2 + y2 – 4y – 6 = 0
x2 + (mx – 8)2 – 4(mx – 8) – 6 = 0
x2 + m2x2 – 16mx + 64 –4mx + 32 – 6 = 0
(m2+ 1)x2 – 20mx + 90 = 0
-8
In this quadratic a = (m2+ 1)
b = -20m
c =90
28. Tangency
Higher
Outcome 4
www.mathsrevision.com
For tangency we need discriminate = 0
b2 – 4ac = 0
(-20m)2 – 4 X (m2+ 1) X 90 = 0
400m2 – 360m2 – 360 = 0
40m2 – 360 = 0
40m2 = 360
m2 = 9
So the two tangents are
m = -3 or 3
y = -3x – 8 and y = 3x - 8
and the gradients are reflected in the symmetry of the diagram.
29. Equations of Tangents
Outcome 4
www.mathsrevision.com
Higher
NB:
At the point of contact
a tangent and radius/diameter are
perpendicular.
Tangent
radius
This means we make use of
m1m2 = -1.
30. Equations of Tangents
www.mathsrevision.com
Higher
Example
Outcome 4
Prove that the point (-4,4) lies on the circle
x 2 + y2 – 12y + 16 = 0
Find the equation of the tangent here.
At (-4,4)
NAB
x2 + y2 – 12y + 16 = 16 + 16 – 48 + 16 = 0
So (-4,4) must lie on the circle.
x2 + y2 – 12y + 16 = 0
2g = 0 so g = 0
2f = -12 so f = -6
Centre is (-g,-f) = (0,6)
31. Equations of Tangents
Outcome 4
www.mathsrevision.com
Higher
(0,6)
Gradient of radius =
y2 – y 1
=
x2 – x 1
=
=
(-4,4)
2
/(0 + 4)
(6 – 4)
/4
1
/2
So gradient of tangent = -2
Using
y – b = m(x – a)
We get
y – 4 = -2(x + 4)
y – 4 = -2x - 8
y = -2x - 4
( m1m2 = -1)
35. Maths4Scotland
Higher
Find the equation of the circle with centre
(–3, 4) and passing through the origin.
Find radius (distance formula):
You know the centre:
Write down equation:
r =5
(−3, 4)
( x + 3) 2 + ( y − 4) 2 = 25
Hint
Previous
Quit
Quit
Next
36. Maths4Scotland
Higher
Explain why the equation
x2 + y2 + 2x + 3 y + 5 = 0
does not represent a circle.
Consider the 2 conditions
1. Coefficients of x2 and y2 must be the same.
2. Radius must be > 0
g +
Evaluate f − c
2
Deduction:
3
f =−
2
g = −1,
Calculate g and f:
2
i.e. g 2 + f 2 − c > 0
(−1) +
2
g 2 + f 2 − c < 0 so
( )
3
−
2
2
−5
1
4
⇒ 1 + 2 −5 < 0
g 2 + f 2 − c not real
Equation does not represent a circle
Hint
Previous
Quit
Quit
Next
37. Maths4Scotland
Higher
Find the equation of the circle which has P(–2, –1) and Q(4, 5)
as the end points of a diameter.
Q(4, 5)
C
Make a sketch
P(-2, -1)
(1, 2)
Calculate mid-point for centre:
Calculate radius CQ:
Write down equation;
r = 18
( x − 1) + ( y − 2 ) = 18
2
2
Hint
Previous
Quit
Quit
Next
38. Maths4Scotland
Higher
Find the equation of the tangent at the point (3, 4) on the circle
x 2 + y 2 + 2 x − 4 y − 15 = 0
Calculate centre of circle:
P(3, 4)
(−1, 2)
Make a sketch
O(-1, 2)
Calculate gradient of OP (radius to tangent)
Gradient of tangent:
1
2
m = −2
Equation of tangent:
m=
y + 2 x = 10
Hint
Previous
Quit
Quit
Next
39. Maths4Scotland
Higher
The point P(2, 3) ( x + 1) + ( y − 1) = 13
lies on the circle
2
2
Find the equation of the tangent at P.
Find centre of circle:
P(2, 3)
(−1, 1)
Make a sketch
O(-1, 1)
Calculate gradient of radius to tangent
Gradient of tangent:
3
m=−
2
Equation of tangent:
m=
2
3
2 y + 3 x = 12
Hint
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40. Maths4Scotland
Higher
O, A and B are the centres of the three circles shown in
the diagram. The two outer circles are congruent, each
2
2
touches thexsmallest circle. )Circle centre A has equation
( − 12 ) + ( y + 5 = 25
The three centres lie on a parabola whose axis of symmetry
is shown the by broken line through A.
y = px( x + q )
a) i) State coordinates of A and find length of line OA.
A(12, of the Find OA (Distance
A is centre of Hencecircle the equation− 5) circle with centre B. formula)
ii) small find
b) The equation of
Find the form
B(24, can
Use symmetry, find B the parabola0) be written inradius of circle A from eqn.
Find radius of circle B
Points O, A, B lie on parabola
– subst. A and B in turn
Previous
13 − 5 = 8
0 = 24 p(24 + q )
−5 = 12 p (12 + q)
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Eqn. of B
Solve:
Find p and q.
13
5
( x − 24) 2 + y 2 = 64
p=
5
,
144
q = −24
Hint
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41. Maths4Scotland
Higher
Circle P has equation x 2 + y 2 − 8 x − 10 y + 9 = 0 Circle Q has centre (–2, –1) and radius 2√2.
a) i) Show that the radius of circle P is 4√2
ii) Hence show that circles P and Q touch.
b) Find the equation of the tangent to circle Q at the point (–4, 1)
a ±b 3
c) The tangent in (b) intersects circle P in two points. Find the x co-ordinates of the points of
Find centre of circle P: (4, 5)
Find radius of circle :P:
42 form2 − 9 = 32 = 4 2
intersection, expressing your answers in the + 5
Find distance between centres
72 = 6 2
Gradient of radius of Q to tangent:
Equation of tangent:
m = −1
Deduction:
Gradient tangent at Q:
m =1
y = x+5
2
2
Solve eqns. simultaneously x + y − 8 x − 10 y + 9 = 0
y = x+5
Previous
= sum of radii, so circles touch
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Soln:
2±2 3
Hint
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42. Maths4Scotland
Higher
For what range of values of k does the equation
represent a circle ?
g = −2k ,
Determine g, f and c:
State condition
g + f −c > 0
2
2
5k 2 + k + 2 > 0
Simplify
(
1
5
)
(
1
5 k +
10
(
5 k
1
+
10
)
)
2
2
Previous
−
1
+2
100
195
+
100
c = −k − 2
Put in values
(−2k ) 2 + k 2 − (−k − 2) > 0
Need to see the position
of the parabola
Complete the square
5 k2 + k + 2
f = k,
x 2 + y 2 + 4kx − 2ky − k − 2 = 0
Minimum value is
195
1
when k = −
100
10
This is positive, so graph is:
Expression is positive for all k:
So equation is a circle for all values of k.
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Hint
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43. Maths4Scotland
Higher
2
2
For what range of values of c does the equation x + y − 6 x + 4 y + c = 0
represent a circle ?
Determine g, f and c:
g = 3,
State condition
g2 + f 2 − c > 0
Simplify
f = −2,
c=?
32 + (−2) 2 − c > 0
9+4−c > 0
Re-arrange:
Put in values
c < 13
Hint
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44. Maths4Scotland
Higher
The circle shown has equation 2
( x − 3) + ( y + 2) 2 = 25
Find the equation of the tangent at the point (6,
2).
Calculate centre of circle:
(3, − 2)
Calculate gradient of radius (to tangent)
3
4
Gradient of tangent:
m=−
Equation of tangent:
4
m=
3
4 y + 3 x = 26
Hint
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45. Maths4Scotland
Higher
When newspapers were printed by lithograph, the newsprint had
to run over three rollers, illustrated in the diagram by 3 circles.
The centres A, B and C of the three circles are collinear.
( x + 12) 2 + ( y + 15) 2 = 25 and ( x − 24) 2 + ( y − 12) 2 = 100
The equations of the circumferences of the outer circles are
Find centre and Find the equation of the ( −12,circle.
radius of Circle A
central − 15)
(24, 12)
Find centre and radius of Circle C
Find diameter of circle B
45 − (5 + 10) = 30
Use proportion to find B
25
× 27
45
Previous
(4, − 3)
r =5
25
r = 10
= 15,
Equation of B
Quit
27
B
20
362 + 27 2 = 45
Find distance AB (distance formula)
Centre of B
(24, 12)
(-12, -15)
36
so radius of B = 15
25
× 36 =
45
20
relative to C
( x − 4 ) + ( y + 3) = 225
2
Quit
2
Next
Hint
We start by find the equation of a circle centre the origin.
First draw set axises x,y and then label the origin O.
Next we plot a point P say, which as coordinates x,y.
Next draw a line from the origin O to the point P and label
length of this line r.
If we now rotate the point P through 360 degrees keep the Origin fixed we trace out a circle with radius r and centre O.
Remembering Pythagoras’s Theorem from Standard grade
a square plus b squared equal c squares we can now write down the equal of any circle with centre the origin.
We are now in a position to find the equation of any circle with centre A,B.
All we have to do is repeat the process in shown in slide 2, but this time the centre is chosen to be (a,b).
First plot a point C and label it’s coordinates (a,b), next we plot another point P and label it’s coordinates (x,y).
Next draw a line from C to P and call this length (r).
(r) will be the radius of our circle with centre (a,b).
Again we rotate the point P through 360 degrees keeping the point C fixed.
Using Pythagoras Theorem a squared plus b squared equal c squared we can write down the equation of any circle with centre (a,b) and radius (r).
The equation is (x - a) all squared plus (y-b) all squared equals (r) squared.
Finally to write down the equation of a circle we need to know the co-ordinates of the centre and the length of the radius or co-ordinates of the centre and the co-ordinates of a point on the circumference of the circle.
We have derived the general equation of any circle with centre (a,b) and radius ®, this is given at the top of the slide.
We can re-write this equation into a different format given at the bottom of the slide.
The reason for doing so, is that this format can be much more useful when dealing with certain types of questions.
To get to this equivalent form we multiply out the bracket in our original equation.
Then we gather each of the different terms together.
Then we equate the LHS of the equation to zero, by subtracting r squared from each side.
We then we tidy up the equation further by letting g= - a , f = -b and the constant term c = a squared plus b squared minus r squared.
This gives us the equivalent form of the general equation at the bottom of the slide.
Note that in this format the centre is given by (-g, -f ) since a=-g and b=-f.
Also by rearranging the expression for c we can deduce the formula for the radius r.
r=square root of g2+f2-c.