The document discusses conic sections including the ellipse and hyperbola. It provides insights beyond the syllabus, including the historical origins of conic sections from Apollonius of Perga's work and their importance in Kepler's laws of planetary motion. The key properties of ellipses and hyperbolas are defined geometrically using focus, directrix, and eccentricity. Their Cartesian and parametric equations are also derived and explained.
The polar coordinate system uses a point called the pole and a fixed ray called the polar axis to identify the location of a point P using polar coordinates (r, θ). R represents the distance from the pole to point P, while θ is the angle between the polar axis and a line extending from the pole to point P. Equations in polar coordinates take the form of r = k sinθ or r = j cosθ, where r is the distance and θ is the angle.
The document describes polar coordinates, which specify the location of a point P in a plane using two numbers: r, the distance from P to the origin O, and θ, the angle between the x-axis and a line from O to P measured counterclockwise. Conversion formulas between polar (r, θ) and rectangular (x, y) coordinates are provided. An example problem converts several polar coordinates to rectangular form and plots the points on a graph.
The document discusses polar coordinates and graphs. It begins by explaining how polar coordinates (r, θ) track the location of a point P in the plane, where r is the distance from the origin and θ is the angle from the positive x-axis. It then provides the conversions between rectangular (x, y) and polar coordinates. The document introduces polar equations, showing how they relate r and θ, and gives examples of the constant equations r = c and θ = c, which describe a circle and line, respectively. It concludes by explaining how to graph other polar equations by plotting points using a polar graph paper.
Futher pure mathematics 3 hyperbolic functionsDennis Almeida
This document provides an overview of hyperbolic functions including:
- Their definition in terms of exponential functions compared to circular/trigonometric functions defined using the unit circle.
- Graphs and properties of the six main hyperbolic functions (sinh, cosh, tanh, sech, coth, cosech) derived using exponential definitions and relationships between functions.
- Typical session structure includes introducing hyperbolic functions, defining the six functions, proving identities, and example exam questions.
The document defines conic sections and describes parabolas. It provides specific objectives related to defining conic sections, identifying different types, describing parabolas, and converting between general and standard forms of parabola equations. It then gives details on the focus, directrix, vertex, latus rectum, and eccentricity of parabolas. Examples of problems involving finding parabola equations and properties from conditions are also provided.
The document discusses parabolas, including their key properties and equations. It defines a parabola as the set of points equidistant from a fixed point (the focus) and a fixed line (the directrix). The document derives the standard equation of a parabola from this definition and discusses how to graph parabolas based on their equations. It also covers transformations, latus rectum, and other geometric properties of parabolas.
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.
The polar coordinate system uses a point called the pole and a fixed ray called the polar axis to identify the location of a point P using polar coordinates (r, θ). R represents the distance from the pole to point P, while θ is the angle between the polar axis and a line extending from the pole to point P. Equations in polar coordinates take the form of r = k sinθ or r = j cosθ, where r is the distance and θ is the angle.
The document describes polar coordinates, which specify the location of a point P in a plane using two numbers: r, the distance from P to the origin O, and θ, the angle between the x-axis and a line from O to P measured counterclockwise. Conversion formulas between polar (r, θ) and rectangular (x, y) coordinates are provided. An example problem converts several polar coordinates to rectangular form and plots the points on a graph.
The document discusses polar coordinates and graphs. It begins by explaining how polar coordinates (r, θ) track the location of a point P in the plane, where r is the distance from the origin and θ is the angle from the positive x-axis. It then provides the conversions between rectangular (x, y) and polar coordinates. The document introduces polar equations, showing how they relate r and θ, and gives examples of the constant equations r = c and θ = c, which describe a circle and line, respectively. It concludes by explaining how to graph other polar equations by plotting points using a polar graph paper.
Futher pure mathematics 3 hyperbolic functionsDennis Almeida
This document provides an overview of hyperbolic functions including:
- Their definition in terms of exponential functions compared to circular/trigonometric functions defined using the unit circle.
- Graphs and properties of the six main hyperbolic functions (sinh, cosh, tanh, sech, coth, cosech) derived using exponential definitions and relationships between functions.
- Typical session structure includes introducing hyperbolic functions, defining the six functions, proving identities, and example exam questions.
The document defines conic sections and describes parabolas. It provides specific objectives related to defining conic sections, identifying different types, describing parabolas, and converting between general and standard forms of parabola equations. It then gives details on the focus, directrix, vertex, latus rectum, and eccentricity of parabolas. Examples of problems involving finding parabola equations and properties from conditions are also provided.
The document discusses parabolas, including their key properties and equations. It defines a parabola as the set of points equidistant from a fixed point (the focus) and a fixed line (the directrix). The document derives the standard equation of a parabola from this definition and discusses how to graph parabolas based on their equations. It also covers transformations, latus rectum, and other geometric properties of parabolas.
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.
The document defines and describes ellipses. It states that an ellipse is the set of points whose sum of the distances to two fixed foci is a constant. An ellipse has a center, major axis, and minor axis. The standard form of the ellipse equation is given as (x-h)2/a2 + (y-k)2/b2 = 1, where (h,k) is the center, a is the x-radius, and b is the y-radius. An example problem demonstrates how to identify these properties from a given ellipse equation and sketch the ellipse.
Conic sections and introduction to circlesArric Tan
Conic sections are shapes that result from slicing a cone with a plane. There are four types of conic sections: circles, ellipses, parabolas, and hyperbolas. Circles can be defined by the general formula x^2 + y^2 = r^2, where all points are a distance r from the center. The center and radius of a circle can be determined by shifting the circle and setting the x and y components to 0.
The document discusses parabolas and their key properties. It defines a parabola as the set of all points equidistant from a fixed point (the focus) and a line (the directrix). The line perpendicular to the directrix through the focus is called the axis. The point where the parabola intersects the axis is the vertex. Parabolas have reflecting properties such that light rays entering parallel to the axis will exit through the focus. This allows parabolic dishes to focus collected radio waves or light at a single point. The document also provides examples of graphing parabolas and finding their equations based on given properties, as well as calculating dimensions of the Hubble Space Telescope's parabolic mirror.
This document defines and provides the standard forms of conic sections, including circles, ellipses, parabolas, and hyperbolas. It explains that a circle is a closed loop where each point is a fixed distance from the center. A parabola is the set of points equidistant from a directrix and focus. An ellipse is the set of points where the sum of distances to two foci is constant. A hyperbola is the set of points where the difference between distances to two foci is constant. Standard forms are provided for each conic section with the vertex or center at (0,0) or (h,k) and characteristics like vertices and foci.
The document discusses polar coordinates and graphs. It begins by explaining how polar coordinates (r, θ) track the location of a point P in the plane, where r is the distance from the origin and θ is the angle from the x-axis. It then provides the conversions between rectangular (x, y) and polar coordinates. The document gives examples of basic polar graphs for constant equations like r = c, which describes a circle, and θ = c, which describes a line. It concludes by explaining how to graph other polar equations using a polar graph paper.
This document discusses conic sections, including circles, ellipses, parabolas, and hyperbolas. It provides:
1) The definitions and standard equations of each conic section, describing how they are formed from the intersection of a plane with a double cone.
2) Examples of different forms the equations can take and the geometric properties of each conic section, such as foci, axes, vertices, and asymptotes.
3) Methods for writing the equations of tangents to conics and using parametric equations to represent loci.
In less than 3 sentences, it summarizes the key information about conic sections provided in the document.
The document discusses calculating the area swept out by a polar function r=f(θ) between two angles θ=A and θ=B. It introduces the integral formula for finding this area, which is ∫f(θ)2dθ from A to B. It then provides examples of using this formula to calculate the areas of different polar curves, such as a circle, a cardioid, and a curve tracing a circle twice.
This document provides information about polar coordinates including:
- Relations between Cartesian and polar coordinates
- Sketching graphs in polar coordinates such as circles, cardioids, and roses
- Finding intersections of curves, slopes of tangents, and areas bounded by polar curves
- Computing arc lengths and surfaces of revolution generated by polar curves
It discusses key concepts like symmetry properties and provides examples of computing specific values related to polar curves.
Polar coordinates provide an alternative way to specify the location of a point P in a plane using two numbers: r, the distance from P to the origin O, and θ, the angle between the x-axis and a line from O to P. θ is measured counter-clockwise from the x-axis and can be either positive or negative. A point P's polar coordinates (r, θ) uniquely identify its location. Polar and rectangular coordinates can be converted between each using the relationships x=r*cos(θ), y=r*sin(θ), and r=√(x2+y2).
The document describes polar coordinates. Polar coordinates represent the location of a point P in a plane using two numbers: r, the distance from P to the origin O, and θ, the angle between the positive x-axis and a line from O to P. θ is positive for counter-clockwise angles and negative for clockwise angles. The polar coordinate (r, θ) uniquely identifies the point P. Conversions between polar coordinates (r, θ) and rectangular coordinates (x, y) are given by the equations x=r*cos(θ), y=r*sin(θ), and r=√(x2+y2).
This document contains 12 theorems regarding circles:
1. Equal chords of a circle subtend equal angles at the centre.
2. If the angles subtended by the chords of a circle at the centre are equal, then the chords are equal.
3. The perpendicular from the centre of a circle to a chord bisects the chord.
This document discusses conic sections and first degree equations. It begins by introducing conic sections as the shapes formed by slicing a cone at different angles. It then covers first degree equations, noting that their graphs are straight lines that can be written in the form of y=mx+b. Specific examples of first degree equations and their graphs are shown. The document ends by introducing the four types of conic sections - circles, ellipses, parabolas, and hyperbolas - and how graphs of second degree equations can represent these shapes.
The document discusses parabolas including their parts, graphs, and equations. It defines a parabola as the locus of points where the distance to the focus is equal to the distance to the directrix. The parts of a parabola include the vertex, focus, directrix, axis of symmetry, and latus rectum. The document outlines the graphs of parabolas with the vertex at the origin or at a point (h,k), and opening in different directions. It notes equations will be provided for parabolas with the vertex at the origin or (h,k), but does not show the actual equations.
Introduction To Polar Coordinates And Graphseekeeney
Polar coordinates use r and θ instead of x and y, where r represents the radius or distance from a point to the pole, and θ represents the angle between the radius and the polar axis. To convert between rectangular and polar coordinates, r is calculated as the distance from the origin using the Pythagorean theorem, while θ is the angle measured counterclockwise from the polar axis. Polar graphs represent r as a function of θ, with θ as the independent variable instead of x as in rectangular graphs.
35 tangent and arc length in polar coordinatesmath266
The document discusses parametric representations of polar curves. It begins by explaining that a rectangular curve given by y=f(x) can be parametrized as x=t and y=f(t). It then explains that a polar curve given by r=f(θ) can be parametrized as x=f(θ)cos(θ) and y=f(θ)sin(θ). Examples are given of parametrizing the Archimedean spiral r=θ and the cardioid r=1+sin(θ). Formulas are derived for calculating the slope of the tangent line to a polar curve at a given point in terms of r and θ. Examples are worked out for finding the slope
EQUATION OF A PARABOLA FROM THE VERTEX AND DIRECTRIXsumanmathews
What is a parabola? How is it derived from conics?
Watch this presentation to find out.
Here, we learn how a parabola is derived when a plane cuts a cone. We learn that, for a parabola, distance of a point from the focus = distance of the point from the directrix. We solve problems based on this principle and also learn how to calculate equation of the axis and the coordinates of the vertex.
This is useful for grade 11 maths students. This channel has videos for grades 11, 12, engineering maths, nata maths and the GRE QUANT section.
Consider subscribing to my channel for more videos. You can visit my page
https://www.mathmadeeasy.co/lessons
For further help, you can join my classes for grade 11 maths
This document discusses parabolas, including their key features like the vertex, focus, directrix, and axis of symmetry. It provides examples of how to graph parabolas given their standard form equations, both for parabolas with vertices at the origin and for parabolas with other vertices. It also shows how to write the standard form equation of a parabola when given its focus and directrix.
The document discusses polar coordinates and graphs. Polar coordinates (r, θ) can be used to specify the location of a point P by giving the distance r from the origin and the angle θ. Conversion formulas allow changing between polar (r, θ) and rectangular (x, y) coordinates. Polar equations relate r and θ, and common ones like r = c (a circle) and θ = c (a line) are examined. Graphing polar equations involves plotting the r and θ values specified by the equation.
The document provides an overview of Module 1 of an analytic geometry course, which covers conic sections. Lesson 1 focuses specifically on circles. It defines a circle, discusses the standard form of a circular equation, and how to graph circles. It also provides an example of stating the center and radius of a circle given its equation. The objectives are to illustrate different conic sections including circles, define and work with circular equations, and solve problems involving circles.
Astronomy Projects For Calculus And Differential EquationsKatie Robinson
This document provides astronomy projects for calculus and differential equations courses that use real-world astronomical phenomena to teach mathematical concepts. It includes introductions to Kepler's laws of planetary motion and approximations of eccentric anomaly using Bessel functions. Projects are provided for calculus I-III and differential equations courses on topics like the orbits of Mars, Mercury, Halley's Comet, the Mars rover Curiosity, and a star orbiting Sagittarius A*. Instructions are given for instructors to assign the projects along with sample student reports.
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.
The document defines and describes ellipses. It states that an ellipse is the set of points whose sum of the distances to two fixed foci is a constant. An ellipse has a center, major axis, and minor axis. The standard form of the ellipse equation is given as (x-h)2/a2 + (y-k)2/b2 = 1, where (h,k) is the center, a is the x-radius, and b is the y-radius. An example problem demonstrates how to identify these properties from a given ellipse equation and sketch the ellipse.
Conic sections and introduction to circlesArric Tan
Conic sections are shapes that result from slicing a cone with a plane. There are four types of conic sections: circles, ellipses, parabolas, and hyperbolas. Circles can be defined by the general formula x^2 + y^2 = r^2, where all points are a distance r from the center. The center and radius of a circle can be determined by shifting the circle and setting the x and y components to 0.
The document discusses parabolas and their key properties. It defines a parabola as the set of all points equidistant from a fixed point (the focus) and a line (the directrix). The line perpendicular to the directrix through the focus is called the axis. The point where the parabola intersects the axis is the vertex. Parabolas have reflecting properties such that light rays entering parallel to the axis will exit through the focus. This allows parabolic dishes to focus collected radio waves or light at a single point. The document also provides examples of graphing parabolas and finding their equations based on given properties, as well as calculating dimensions of the Hubble Space Telescope's parabolic mirror.
This document defines and provides the standard forms of conic sections, including circles, ellipses, parabolas, and hyperbolas. It explains that a circle is a closed loop where each point is a fixed distance from the center. A parabola is the set of points equidistant from a directrix and focus. An ellipse is the set of points where the sum of distances to two foci is constant. A hyperbola is the set of points where the difference between distances to two foci is constant. Standard forms are provided for each conic section with the vertex or center at (0,0) or (h,k) and characteristics like vertices and foci.
The document discusses polar coordinates and graphs. It begins by explaining how polar coordinates (r, θ) track the location of a point P in the plane, where r is the distance from the origin and θ is the angle from the x-axis. It then provides the conversions between rectangular (x, y) and polar coordinates. The document gives examples of basic polar graphs for constant equations like r = c, which describes a circle, and θ = c, which describes a line. It concludes by explaining how to graph other polar equations using a polar graph paper.
This document discusses conic sections, including circles, ellipses, parabolas, and hyperbolas. It provides:
1) The definitions and standard equations of each conic section, describing how they are formed from the intersection of a plane with a double cone.
2) Examples of different forms the equations can take and the geometric properties of each conic section, such as foci, axes, vertices, and asymptotes.
3) Methods for writing the equations of tangents to conics and using parametric equations to represent loci.
In less than 3 sentences, it summarizes the key information about conic sections provided in the document.
The document discusses calculating the area swept out by a polar function r=f(θ) between two angles θ=A and θ=B. It introduces the integral formula for finding this area, which is ∫f(θ)2dθ from A to B. It then provides examples of using this formula to calculate the areas of different polar curves, such as a circle, a cardioid, and a curve tracing a circle twice.
This document provides information about polar coordinates including:
- Relations between Cartesian and polar coordinates
- Sketching graphs in polar coordinates such as circles, cardioids, and roses
- Finding intersections of curves, slopes of tangents, and areas bounded by polar curves
- Computing arc lengths and surfaces of revolution generated by polar curves
It discusses key concepts like symmetry properties and provides examples of computing specific values related to polar curves.
Polar coordinates provide an alternative way to specify the location of a point P in a plane using two numbers: r, the distance from P to the origin O, and θ, the angle between the x-axis and a line from O to P. θ is measured counter-clockwise from the x-axis and can be either positive or negative. A point P's polar coordinates (r, θ) uniquely identify its location. Polar and rectangular coordinates can be converted between each using the relationships x=r*cos(θ), y=r*sin(θ), and r=√(x2+y2).
The document describes polar coordinates. Polar coordinates represent the location of a point P in a plane using two numbers: r, the distance from P to the origin O, and θ, the angle between the positive x-axis and a line from O to P. θ is positive for counter-clockwise angles and negative for clockwise angles. The polar coordinate (r, θ) uniquely identifies the point P. Conversions between polar coordinates (r, θ) and rectangular coordinates (x, y) are given by the equations x=r*cos(θ), y=r*sin(θ), and r=√(x2+y2).
This document contains 12 theorems regarding circles:
1. Equal chords of a circle subtend equal angles at the centre.
2. If the angles subtended by the chords of a circle at the centre are equal, then the chords are equal.
3. The perpendicular from the centre of a circle to a chord bisects the chord.
This document discusses conic sections and first degree equations. It begins by introducing conic sections as the shapes formed by slicing a cone at different angles. It then covers first degree equations, noting that their graphs are straight lines that can be written in the form of y=mx+b. Specific examples of first degree equations and their graphs are shown. The document ends by introducing the four types of conic sections - circles, ellipses, parabolas, and hyperbolas - and how graphs of second degree equations can represent these shapes.
The document discusses parabolas including their parts, graphs, and equations. It defines a parabola as the locus of points where the distance to the focus is equal to the distance to the directrix. The parts of a parabola include the vertex, focus, directrix, axis of symmetry, and latus rectum. The document outlines the graphs of parabolas with the vertex at the origin or at a point (h,k), and opening in different directions. It notes equations will be provided for parabolas with the vertex at the origin or (h,k), but does not show the actual equations.
Introduction To Polar Coordinates And Graphseekeeney
Polar coordinates use r and θ instead of x and y, where r represents the radius or distance from a point to the pole, and θ represents the angle between the radius and the polar axis. To convert between rectangular and polar coordinates, r is calculated as the distance from the origin using the Pythagorean theorem, while θ is the angle measured counterclockwise from the polar axis. Polar graphs represent r as a function of θ, with θ as the independent variable instead of x as in rectangular graphs.
35 tangent and arc length in polar coordinatesmath266
The document discusses parametric representations of polar curves. It begins by explaining that a rectangular curve given by y=f(x) can be parametrized as x=t and y=f(t). It then explains that a polar curve given by r=f(θ) can be parametrized as x=f(θ)cos(θ) and y=f(θ)sin(θ). Examples are given of parametrizing the Archimedean spiral r=θ and the cardioid r=1+sin(θ). Formulas are derived for calculating the slope of the tangent line to a polar curve at a given point in terms of r and θ. Examples are worked out for finding the slope
EQUATION OF A PARABOLA FROM THE VERTEX AND DIRECTRIXsumanmathews
What is a parabola? How is it derived from conics?
Watch this presentation to find out.
Here, we learn how a parabola is derived when a plane cuts a cone. We learn that, for a parabola, distance of a point from the focus = distance of the point from the directrix. We solve problems based on this principle and also learn how to calculate equation of the axis and the coordinates of the vertex.
This is useful for grade 11 maths students. This channel has videos for grades 11, 12, engineering maths, nata maths and the GRE QUANT section.
Consider subscribing to my channel for more videos. You can visit my page
https://www.mathmadeeasy.co/lessons
For further help, you can join my classes for grade 11 maths
This document discusses parabolas, including their key features like the vertex, focus, directrix, and axis of symmetry. It provides examples of how to graph parabolas given their standard form equations, both for parabolas with vertices at the origin and for parabolas with other vertices. It also shows how to write the standard form equation of a parabola when given its focus and directrix.
The document discusses polar coordinates and graphs. Polar coordinates (r, θ) can be used to specify the location of a point P by giving the distance r from the origin and the angle θ. Conversion formulas allow changing between polar (r, θ) and rectangular (x, y) coordinates. Polar equations relate r and θ, and common ones like r = c (a circle) and θ = c (a line) are examined. Graphing polar equations involves plotting the r and θ values specified by the equation.
The document provides an overview of Module 1 of an analytic geometry course, which covers conic sections. Lesson 1 focuses specifically on circles. It defines a circle, discusses the standard form of a circular equation, and how to graph circles. It also provides an example of stating the center and radius of a circle given its equation. The objectives are to illustrate different conic sections including circles, define and work with circular equations, and solve problems involving circles.
Astronomy Projects For Calculus And Differential EquationsKatie Robinson
This document provides astronomy projects for calculus and differential equations courses that use real-world astronomical phenomena to teach mathematical concepts. It includes introductions to Kepler's laws of planetary motion and approximations of eccentric anomaly using Bessel functions. Projects are provided for calculus I-III and differential equations courses on topics like the orbits of Mars, Mercury, Halley's Comet, the Mars rover Curiosity, and a star orbiting Sagittarius A*. Instructions are given for instructors to assign the projects along with sample student reports.
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.
An ellipse is a closed curve where the sum of the distances from two fixed points (foci) to any point on the curve is a constant. It can be defined parametrically using angles or implicitly as a second-degree equation. Key properties include the major and minor axes, which are lines of symmetry, and the eccentricity, which is a measure of how non-circular the ellipse is. The area of an ellipse is πab and its circumference can be calculated using elliptic integrals.
The Kaybiang Tunnel is the longest elliptical shaped tunnel in the Philippines, connecting two towns. An ellipse is the set of all points in a plane where the sum of the distances from two fixed points, called foci, is a constant. The Kaybiang Tunnel has an elliptical shape with its longest section piercing through a mountain.
This document contains examples and explanations about lines and circles. It defines key terms like chord, secant, tangent, diameter, and radius. It then provides examples of identifying these features when they intersect circles. Subsequent examples show finding radii, points of tangency, and writing equations of tangent lines. Other examples demonstrate using properties of tangents to solve problems and find measures of arcs and angles related to circles.
“Conic section” is a fundamental of the Mathematics. This
report is made from my studying about the conic section in the
Mathematics books and on the internet. This report contains
topics that involve with conic section such as: The history of Conic
section studying, Parabola, Ellipse, Hyperbola and their
applications with figures may help you to understand easily.
This report is may use to refer for next time and its can be
usefulness for the readers.
The document discusses geometric and analytical thinking. It begins by defining analytical geometry as the science that combines algebra and geometry to describe geometric figures from both algebraic and geometric viewpoints. It then discusses how analytical geometry originated with René Descartes' use of the Cartesian plane. Several geometric figures are then analyzed, including lines, circles, ellipses, and parabolas. Their key parameters and equations are defined. In particular, it provides the canonical equations for circles, ellipses, and parabolas, and discusses topics like slope and parallelism for lines.
An ellipse is a set of points in a plane where the sum of the distances to two fixed points (foci) is a constant. The two foci and the midpoint between them (the center) define key properties of the ellipse, including its major and minor axes. The standard equation of an ellipse with its center at the origin is (x^2)/a^2 + (y^2)/b^2 = 1, where a and b are the lengths of the semi-major and semi-minor axes. The eccentricity, e, is the ratio of the distance from the center to a focus (c) to the semi-major axis length (a).
This document discusses hyperbolas, ellipses, and canonical equations in analytic geometry. It provides definitions and parameters for hyperbolas and ellipses, including centers, vertices, foci, axes, and asymptotes. Examples are given of determining parameters from equations and graphing conic sections. The document emphasizes using canonical equations to solve problems involving hyperbolas and ellipses. It also outlines steps for solving problems using analytic geometry, such as identifying the relevant figure and equations.
Paso 4_Álgebra, trigonometría y Geometría AnalíticaTrigogeogebraunad
The document discusses different types of conic sections including circles, ellipses, hyperbolas, and parabolas. It provides the general equation for conic sections and the conditions to determine which type of conic section is represented based on the values of certain coefficients in the equation. It then gives the standard forms of the equations for each type of conic section and discusses some of their defining geometric properties.
1) The document discusses the electronic configuration of atoms, including the development of wave mechanics and quantum theory to explain the structure of atoms. It introduces concepts like the de Broglie wavelength, quantum numbers, atomic orbitals and shapes, Pauli's exclusion principle, and Hund's rule for electron configuration.
2) Key scientists discussed include de Broglie, Heisenberg, Schrodinger, Pauli, and their contributions to developing models of the atom and allowing prediction of electron configurations.
3) The document provides examples of writing out electron configurations for elements and explaining the rules for filling atomic orbitals in the Aufbau principle.
The document discusses various conic sections including circles, parabolas, ellipses, and hyperbolas. It provides definitions, key properties, and standard equations for each type of conic section. Examples of applications in fields like planetary motion and antenna design are also mentioned. Key aspects like focus, directrix, eccentricity, vertex and axis are defined for parabolas, ellipses, and hyperbolas. Methods to find characteristics like latus rectum, equation from geometric properties are also demonstrated.
René Descartes introduced innovative algebraic techniques for analyzing geometric problems and understanding the connection between a curve's construction and algebraic equation in his work La Géométrie
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 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
Similar to Further pure mathematics 3 coordinate systems (20)
The document describes the Vigenère cipher method of encryption. It involves using a keyword that is repeated above the plaintext message. Each letter of the plaintext is then shifted according to the corresponding letter of the keyword, using a table of letter shifts. The document provides instructions for cracking a Vigenère cipher by finding repeating blocks of letters and using the frequency of letters to deduce the keyword length and decrypt the message.
1. Al-Khwarizmi wrote the first treatise on algebra, Hisab al-jabr w’al-muqabala, in 820 AD, which provided methods for solving equations. The word "algebra" is derived from "al-jabr" in the title, meaning restoration.
2. Pedro Nunes published the first known European translation of Al-Khwarizmi's work in 1567. Nunes demonstrated geometric representations of algebraic concepts like expanding brackets and completing the square.
3. Al-Sijzi proved geometrically in the 10th century that the binomial expansion of (a + b)3 is a3 + 3ab(a + b) +
This document provides an overview of key concepts in vectors and vector algebra covered in an Edexcel FP3 module. It begins with an introduction to vectors and quaternions. Section 1 defines vectors and the vector product. Section 2 defines the triple scalar product and its use in calculating the volume of a tetrahedron. It discusses properties of the vector product, including that it yields a vector perpendicular to the two original vectors and has magnitude equal to the product of the lengths of the original vectors and the sine of the angle between them. It provides examples of using the vector and triple scalar products to calculate the area of a parallelogram and the volume of a tetrahedron.
This document provides an overview of the content that will be covered in a session on teaching matrices for the Edexcel FP3 module. The session will begin with an introduction to matrices, including examples and definitions. It will then cover linear transformations represented by matrices, showing how matrices can represent rotations. Subsequent sections will address the transpose and inverse of matrices, eigenvalues and eigenvectors, and symmetric matrices. The session will conclude with examples of typical examination questions. The overall aim is to provide insights beyond the syllabus to effectively teach the FP3 module on matrices.
Anecdotes from the history of mathematics ways of selling mathematiDennis Almeida
1) The development of mathematics, including number systems and arithmetic, was driven by practical needs in areas like trade, taxation, and military affairs. Place value systems like the Hindu-Arabic numerals made complex calculations possible.
2) Early algebra developed out of solving practical problems involving lengths and areas. Techniques like extracting roots and solving quadratic equations were applied to problems in areas like right triangles and bone setting.
3) Geometry originated from practical construction needs but was formalized by Euclid into a deductive system. It influenced fields like art and tiling patterns. Relating geometric concepts to algebraic formulas helped develop modern algebra.
This lesson plan introduces equivalent fractions and simplifying fractions to a whole class. It aims to teach students to identify equivalent fractions by increasing the partition and to simplify fractions by decreasing the partition. Resources needed include worksheets, textbooks, and calculators. Formative assessment includes oral questions, written questions, and homework.
A pupil actively constructs their own mathematical knowledge by interacting new ideas with existing ideas, which can lead to misconceptions. Diagnostic teaching is important as it involves identifying misconceptions, challenging them through discussion to resolve conflicts, and replacing misconceptions with correct understanding. The teacher must understand the source of the misconception to effectively challenge it, and research shows this diagnostic approach promotes better learning compared to simply explaining again.
The document discusses equivalent fractions and provides examples of finding equivalent fractions by adding the same number to both the numerator and denominator. It also covers ordering fractions by finding a common denominator and writing fractions with the same denominator. Examples are given of ordering fractions and finding missing numerators or denominators of equivalent fractions.
A review of the growth of the Israel Genealogy Research Association Database Collection for the last 12 months. Our collection is now passed the 3 million mark and still growing. See which archives have contributed the most. See the different types of records we have, and which years have had records added. You can also see what we have for the future.
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How to Build a Module in Odoo 17 Using the Scaffold MethodCeline George
Odoo provides an option for creating a module by using a single line command. By using this command the user can make a whole structure of a module. It is very easy for a beginner to make a module. There is no need to make each file manually. This slide will show how to create a module using the scaffold method.
Strategies for Effective Upskilling is a presentation by Chinwendu Peace in a Your Skill Boost Masterclass organisation by the Excellence Foundation for South Sudan on 08th and 09th June 2024 from 1 PM to 3 PM on each day.
The simplified electron and muon model, Oscillating Spacetime: The Foundation...RitikBhardwaj56
Discover the Simplified Electron and Muon Model: A New Wave-Based Approach to Understanding Particles delves into a groundbreaking theory that presents electrons and muons as rotating soliton waves within oscillating spacetime. Geared towards students, researchers, and science buffs, this book breaks down complex ideas into simple explanations. It covers topics such as electron waves, temporal dynamics, and the implications of this model on particle physics. With clear illustrations and easy-to-follow explanations, readers will gain a new outlook on the universe's fundamental nature.
How to Fix the Import Error in the Odoo 17Celine George
An import error occurs when a program fails to import a module or library, disrupting its execution. In languages like Python, this issue arises when the specified module cannot be found or accessed, hindering the program's functionality. Resolving import errors is crucial for maintaining smooth software operation and uninterrupted development processes.
Walmart Business+ and Spark Good for Nonprofits.pdfTechSoup
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This slide is special for master students (MIBS & MIFB) in UUM. Also useful for readers who are interested in the topic of contemporary Islamic banking.
How to Add Chatter in the odoo 17 ERP ModuleCeline George
In Odoo, the chatter is like a chat tool that helps you work together on records. You can leave notes and track things, making it easier to talk with your team and partners. Inside chatter, all communication history, activity, and changes will be displayed.
This presentation was provided by Steph Pollock of The American Psychological Association’s Journals Program, and Damita Snow, of The American Society of Civil Engineers (ASCE), for the initial session of NISO's 2024 Training Series "DEIA in the Scholarly Landscape." Session One: 'Setting Expectations: a DEIA Primer,' was held June 6, 2024.
2. Philosophy of the course
• The course does not aim to present a teaching plan for any of
the FP3 topics. This is left to your professional judgement as a
teacher.
• The course is more about presenting each topic in a way that
provides insights into the topic beyond the syllabus.
• In general the course will present the necessary knowledge and
skills to teach the Edexcel FP3 module effectively.
• Relevant examination problem solving techniques will be
demonstrated.
3. Session structure
• Section 1: Introduction to the ellipse and hyperbola.
• Section 2: Cartesian and parametric equations of the ellipse and
hyperbola.
• Section 3: Properties of the ellipse and hyperbola: eccentricity,
foci and directrices.
• Section 4: Tangents and normals to the ellipse and hyperbola.
• Section 5. Typical examination questions.
5. We get the hyperbola
and the ellipse.
If a double
cone is cut in
a certain way.
The circle, ellipse, parabola and hyperbola are called the conic
sections. This web page http://bit.ly/1dJzJ1Y gives an animation of
how an ellipse might be generated from a cone.
Essentially this topic extends the study of FP1 Coordinate geometry which
introduced some fundamental properties of the parabola and hyperbola. Here we
study the ellipse and the hyperbola to a greater depth.
The ellipse and the hyperbola are from a family of curves called the conic sections.
6. • Apollonius of Perga (approx. 262 BC–190 BC) was a Greek geometer who
studied with Euclid. He is best known for his work on cross sections of a cone.
This is the first recorded study of the conic sections and led to a school of
mathematics which developed the important work of Apollonius.
• Important Arabic additions to the conics from the tenth and eleventh century
were also crucial to the mathematics of the scientific revolution.
• In fact, even to this day, large parts The Conics of Apollonius do not exist in
their original Greek form, and are known to us only through Arabic translations.
• http://www.quadrivium.info/MathInt/Notes/Apollonius.pdf
Abrief history of the conic sections.
7. The importance of the ellipse.
Kepler model of the elliptical orbits of
planets in a solar system →
The ellipse was brought into scientific prominence when Johannes
Kepler proved that planetary orbits were not circular but elliptical.
http://csep10.phys.utk.edu/astr161/lect/history/kepler.html
Keplers laws of planetary motion:
First Law: The orbit of every planet is an ellipse with the sun at one of the
foci.
Second Law: A line joining a planet and the Sun sweeps out equal areas
during equal intervals of time.
Third Law: The square of the orbital period of a planet is directly
proportional to the cube of the semi-major axis of its orbit.
8. Hyperbolae are also important in astronomy:
The majority of comets are
in hyperbolic or parabolic
orbits.
If a comet’s initial velocity is
insignificant the path is
parabolic, whereas if the
initial velocity is significant,
the path is hyperbolic.
http://bit.ly/1f5Xi21
Ellipses and hyperbolae are also important
in other aspects of science and in
economics. You can find more here:
http://bit.ly/1igiw26
10. The ellipse as a transformed circle.
a
O
.
,
value
fixed
by the
circle
on the
points
all
of
coordinate
he
Multiply t
:
way
in this
circle
the
transform
Now
a
b
a
b
P
y
:
0
,
circle
he
Consider t 2
2
2
a
a
y
x
●(x, y)
●
y
a
b
x,
circle.
on the
lies
)
,
(
then
ellipse
on the
lies
)
,
(
If
:
follows
as
found
be
can
ellipse
the
of
equation
The
Y
b
a
X
Y
X
ellipse.
general
the
of
equation
Cartesian
The
1
So
2
2
2
2
2
2
2
b
Y
a
X
a
Y
b
a
X
red.
in
shown
ellipse,
an
is
circle
ed
transform
The
11. Fundamental properties of the ellipse.
A: (a, 0)
O
b
y
b
y
x
y
a
x
a
x
y
x
b
y
a
x
1
0
when
axis
the
intersects
1
0
when
axis
the
intersects
1
equation
Cartesian
with
ellipse
The
2
2
2
2
2
2
2
2
B (–a, 0)
C (0, b)
D (0, –b)
If a > b then AB is called the
major axis and CD is called the
minor axis of the ellipse.
If the other case when b < a,
AB will be the minor and CD
the major axis.
12. Parametric equations of the ellipse.
(a, 0)
O
diagram.
in the
indicated
are
2π
,
2
3π
π,
,
2
π
,
0
with
ellipse
on the
points
The
(1).
equation
Cartesian
with
ellipse
this
of
equations
parametric
the
gives
)
2
(
)
2
(
..........
sin
,
cos
or
sin
,
cos
writing
to
us
leads
This
1
sin
cos
it with
comparing
by
ed
parametris
be
can
)
1
.(
..........
1
equation
Cartesian
with
ellipse
The
2
2
2
2
2
2
t
t
b
y
t
a
x
t
b
y
t
a
x
t
t
b
y
a
x
(0, b)
t = 0
t = 𝟏
𝟐
t =
t = 3
2
t = 2
13. Rectangular to general hyperbolae
hyperbola
ed
transform
the
of
equation
the
:
2
2
are
s
coordinate
new
the
So
matrix
the
by
ed
transform
are
s
coordinate
the
know
we
FP1
From
clockwise.
4
by
graph
this
rotate
we
Now
graph
has
This
FP1.
in
met
hyperbola
r
rectangula
he
Consider t
2
2
2
2
2
2
1
2
2
1
2
2
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
c
Y
X
xy
Y
X
y
x
y
x
Y
X
y
x
y
x
Y
X
y
x
Y
X
c
xy
↓ rotate ¼
clockwise
Note: x = 0 and y = 0
are lines of symmetry
14. Rectangular to general hyperbolae
c
Y
X
c
xy 2
2
2
Perpendicular
asymptotes.
Perpendicular
asymptotes.
later.
verified
be
will
This
:
1
is
0
and
0
symmetry
of
lines
with
hyperbolae
general
the
of
equation
Cartesian
The
asymptotes
lar
perpendicu
have
not
do
hyperbolae
general
In
2
2
2
2
b
y
a
x
y
x
15. Properties and parametric equations of the hyperbola.
)
1
....(
tan
,
sec
s
coordinate
parametric
implies
1
tan
sec
identity
The
).
0
,
(
and
)
0
,
(
at
axis
Intersects
1
2
2
2
2
2
2
t
b
y
t
a
x
t
t
a
a
x
b
y
a
x
(– a, 0) (a, 0)
t = 0
t =
)
2
....(
sinh
,
cosh
use
we
,
For
.
1
cosh
because
when
applies
only
But this
)
2
....(
sinh
,
cosh
s
coordinate
parametric
implies
1
sinh
cosh
and
1
Also 2
2
2
2
2
2
a
t
b
y
t
a
x
a
x
t
a
x
t
b
y
t
a
x
t
t
b
y
a
x
(a, 0)
t = 0
16. Exercises: Equations of the ellipse and hyperbola
functions.
tric
trigonome
of
in terms
form
parametric
the
Give
.
graph.
sketch
a
Provide
.
1
form
equivalent
the
Give
:
above
equations
the
of
each
For
144
16
36
.
4
36
9
4
.
3
36
4
9
.
2
16
4
.
1
2
2
2
2
2
2
2
2
2
2
2
2
iii
ii
b
y
a
x
i.
y
x
y
x
y
x
y
x
17. Exercises: Equations of the ellipse and hyperbola (solutions)
.
sin
2
,
cos
4
:
is
form
parametric
The
.
:
is
graph
sketch
A
.
1
2
4
16
4
16
4
.
1
2
2
2
2
2
2
2
2
y
x
iii
ii
y
x
y
x
i.
y
x
4
– 4
– 2
2
.
sin
3
,
cos
2
:
is
form
parametric
The
.
:
is
graph
sketch
A
.
1
3
2
36
4
9
36
4
9
.
2
2
2
2
2
2
2
2
2
y
x
iii
ii
y
x
y
x
i.
y
x
2
– 2
– 3
3
.
tan
2
,
sec
3
:
is
form
parametric
The
.
:
is
graph
sketch
A
.
1
2
3
36
9
4
36
9
4
.
3
2
2
2
2
2
2
2
2
y
x
iii
ii
y
x
y
x
i.
y
x
3
– 3
.
tan
3
,
sec
2
:
is
form
parametric
The
.
:
is
graph
sketch
A
.
1
3
2
144
16
36
144
16
36
.
4
2
2
2
2
2
2
2
2
y
x
iii
ii
y
x
y
x
i.
y
x
2
– 2
18. •Section 3: Properties of the
ellipse and hyperbola:
eccentricity, foci and directrices.
19. Geometrical derivation of the ellipse.
•One way (but not the only one) to define the conic sections -
parabola, hyperbola and ellipse – is the following :
•They are the loci of points P in the plane that satisfy the
following condition:
•The distance of P from a fixed point S = e the distance
of P from a fixed line L.
•The fixed point S is called the focus of the conic section.
•The fixed line L is called its directrix.
•The constant multiple e is called its eccentricity.
• e =1 for the parabola, 0 < e < 1 for the ellipse and e > 1 for the
hyperbola.
20. Focus, directrix and Cartesian equation of the parabola.
● S (a, 0)
L: x = – a
Let the focus be the point S (a, 0)
and the directrix L: x = – a.
Points P (x, y) on the parabola
satisfy: PS = 1PT.
● P (x, y)
T (– a , y)●
Or PS2 = PT2
ax
y
ax
a
x
y
ax
a
x
a
x
y
a
x
4
2
2
)
(
)
(
2
2
2
2
2
2
2
2
2
21. Focus, directrix and Cartesian equation of the ellipse.
● S (ae, 0)
L: x = a/e
Here we let the focus be the point S
(ae, 0) and the directrix
L: x = a/e, where 0 < e < 1, is the
eccentricity of the ellipse.
Points P (x, y) on the ellipse
satisfy: PS = ePT : 0 < e < 1
● P (x, y) ● T (a/e , y)
Or PS2 = e2PT2
)
1
(
where
,
1
or
1
)
1
(
)
1
(
)
1
(
2
2
)
(
)
(
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
e
a
b
b
y
a
x
e
a
y
a
x
e
a
y
e
x
aex
a
x
e
y
aex
e
a
x
x
e
a
e
y
ae
x
22. Focus, directrix and Cartesian equation of the ellipse.
● S (ae, 0)
L: x = a/e
As the equation of the ellipse has even
powers of x and of y it is symmetrical
about both axes.
This symmetry implies that the ellipse
has two foci and two directrices.
The other focus is S′ (–ae, 0) and other
directrix is L′ : x = –a/e.
● P (x, y) ● T (a/e , y)
.
1
0
),
1
(
where
,
1
is
ellipse
the
of
equation
general
The
2
2
2
2
2
2
2
e
e
a
b
b
y
a
x
L′ : x = –a/e
S′ (–ae, 0) ●
)
0
,
(
and
)
0
,
(
are
intercepts
the
So
.
1
0
)
0
,
(
and
)
0
,
(
are
intercepts
the
So
.
1
0
2
2
2
2
b
b
y
b
y
b
y
x
a
a
x
a
x
a
x
y
(a, 0)
(–a, 0)
(–b, 0)
(b, 0)
23. Focus, directrix and Cartesian equation of the hyperbola.
● S (ae, 0)
L: x = a/e
Here we let the focus be the point S
(ae, 0) and the directrix
L: x = a/e, where e, e >1, is the
eccentricity of the hyperbola.
Points P (x, y) on the hyperbola
satisfy: PS = ePT : e > 1
● P (x, y)
T (a/e , y)●
Or PS2 = e2PT2
0
ensure
to
:
)
1
(
write
to
need
t we
reason tha
the
Note
)
1
(
where
,
1
or
1
)
1
(
)
1
(
)
1
(
)
1
(
)
1
(
2
2
)
(
)
(
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
b
e
e
a
b
b
y
a
x
e
a
y
a
x
e
a
y
e
x
e
a
y
e
x
aex
a
x
e
y
aex
e
a
x
e
a
x
e
y
ae
x
24. Focus, directrix and Cartesian equation of the hyperbola.
Again the equation of the hyperbola has
even powers of x and of y it is
symmetrical about both axes.
The symmetry implies that it has two
foci and two directrices.
The other focus is S′ (–ae, 0) and other
directrix is L′ : x = –a/e.
.
1
),
1
(
where
,
1
is
hyperbola
the
of
equation
general
The
2
2
2
2
2
2
2
e
e
a
b
b
y
a
x
.
intercepts
no
Hence
.
impossible
is
which
,
1
0
)
0
,
(
and
)
0
,
(
are
intercepts
the
So
.
1
0
2
2
2
2
y
b
y
x
a
a
x
a
x
a
x
y
(a, 0)● ● S (ae, 0)
L: x = a/e
T (a/e , y)●
L′ : x = –a/e
S′ (–ae, 0) ●
● P (x, y)
(–a, 0)
●
25. Example: Focus and directrix of the ellipse.
12
12
)
4
1
1
(
16
),
1
(
As
4
(1)
Then
)
ellipse
the
for
1
0
(
2
1
4
1
2
8
)
2
(
and
)
1
(
)
2
(
..........
2
So
).
0
,
(
0)
(2,
focus
The
)
1
(
..........
8
So
.
8
directrix
The
:
2
2
2
2
2
b
b
e
a
b
a
e
e
e
e
e
e
a
ae
e
a
e
a
x
Solution
26. Example: Focus and directrix of the hyperbola.
5
16
are
s
directrice
The
)
0
,
5
(
)
0
,
(
are
foci
The
hyperbola)
for the
1
(
4
5
16
25
16
9
)
1
(
)
1
(
16
9
So
)
1
(
hyperbola
For the
.
3
and
4
get
we
1
form
standard
with the
1
9
16
Comparing
.
directices
and
foci
the
showing
hyperbola
Sketch the
s.
directrice
its
of
equations
the
and
foci
its
of
s
coordinate
the
Find
1
9
16
equation
has
hyperbola
A
2
2
2
2
2
2
2
2
2
2
2
2
2
2
e
a
x
ae
e
e
e
e
e
e
a
b
b
a
b
y
a
x
y
x
Solution:
y
x
●
S (5, 0)
S ′(– 5, 0)
●
x = 3.2
x = – 3.2
27. Exercises: Focus and directrix
above.
sections
conic
the
of
each
for
s
directrice
and
foci
the
Find
1
7
16
.
3
1
3
4
.
2
1
3
4
.
1
2
2
2
2
2
2
y
x
y
x
y
x
28. Exercises: Focus and directrix (solutions)
3
16
s
Directrice
);
0
,
3
(
)
0
,
(
Foci
.
4
3
.
16
9
)
1
(
16
7
)
1
(
.
7
,
16
ellipse.
An
.
1
7
16
.
3
7
7
4
7
4
s
Directrice
);
0
,
7
(
)
0
,
(
Foci
.
2
7
.
4
7
)
1
(
4
3
so
),
1
(
Here
.
3
,
4
1
:
hyperbola
a
of
equation
the
is
This
.
1
3
4
.
2
4
s
Directrice
);
0
,
1
(
)
0
,
(
Foci
.
2
1
.
4
1
)
1
(
4
3
so
),
1
(
Here
.
3
,
4
1
:
ellipse
an
of
equation
the
is
This
.
1
3
4
.
1
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
e
a
x
ae
e
e
e
e
a
b
b
a
y
x
e
a
x
ae
e
e
e
e
a
b
b
a
b
y
a
x
y
x
e
a
x
ae
e
e
e
e
a
b
b
a
b
y
a
x
y
x
30. The gradient function of the ellipse.
(1)}
in
s
coordinate
parametric
the
subs
by
(1)
from
derived
be
can
(2)
{n.b.
)
2
.........(
sin
3
cos
2
cos
2
,
sin
2
sin
2
,
cos
3
are
1
4
9
of
s
coordinate
parametric
The
:
ation
differenti
parametric
By
2.
)
1
(
..........
9
4
0
9
4
0
4
2
9
2
1
4
9
:
ation
differenti
implicit
By
1.
:
ways
in two
determined
be
can
function
gradient
The
.
1
4
9
ellipse
he
Consider t
2
2
2
2
2
2
t
t
dx
dy
dt
dx
dt
dy
dx
dy
t
dt
dy
t
dt
dx
t
y
t
x
y
x
y
x
dx
dy
dx
dy
y
x
dx
dy
y
x
y
x
y
x
31. The gradient function of the ellipse generalised.
)
2
.........(
sin
cos
)
1
(
..........
.
1
ellipse
For the
:
general
made
be
easily
can
slide
previous
the
of
n
calculatio
The
2
2
2
2
2
2
t
a
t
b
dx
dy
y
a
x
b
dx
dy
b
y
a
x
32. The equations of the tangent and normal of the ellipse.
before)
as
(
3
2
sin
3
cos
2
sin
3
cos
2
tangent
the
of
gradient
Then the
n
calculatio
or
inspection
By
).
,
(
)
sin
2
,
cos
3
(
in
find
to
need
tion we
differenta
parametric
using
gradient
the
find
To
.
n.b
0
cos
6
sin
6
2
3
or
cos
6
2
sin
6
3
)
cos
3
(
2
)
sin
2
(
3
)
cos
3
(
3
2
)
sin
2
(
:
is
tangent
the
of
equation
an
So
)
sin
2
,
cos
3
(
are
s
coordinate
parametric
The
:
form
parametric
In
2.
0
2
6
2
3
2
3
)
(
2
)
(
3
)
(
3
2
)
(
:
is
tangent
the
of
equation
an
So
3
2
9
4
9
4
tangent
the
of
gradient
The
:
form
Cartesian
In
1.
:
ways
in two
determined
be
can
)
,
(
at
ellipse
this
o
tangent t
the
of
equation
An
.
1
4
9
ellipse
on the
)
,
(
point
he
Consider t
4
4
4
2
2
2
2
2
3
2
2
6
2
2
6
2
2
3
2
2
2
2
2
3
2
2
2
2
2
2
2
2
3
2
2
2
2
2
3
2
2
2
2
2
2
2
3
t
t
dx
dy
t
t
t
t
t
t
x
y
t
x
t
y
t
x
t
y
t
x
t
y
t
t
x
y
x
y
x
y
x
y
y
x
dx
dy
y
x
33. The gradient function of the hyperbola
(1)}
in
s
coordinate
parametric
relevant
the
subs
by
(1)
from
derived
be
can
(3)
and
(2)
{n.b.
)
3
.........(
sinh
3
cosh
2
cosh
2
2
2
,
sinh
3
2
3
2
2
sinh
2
,
2
3
cosh
3
are
here
s
coordinate
parametric
The
:
ation
differenti
parametric
By
2b.
)
2
.........(
tan
3
sec
2
tan
sec
3
sec
2
sec
2
,
tan
sec
3
tan
2
,
sec
3
are
here
s
coordinate
parametric
The
:
ation
differenti
parametric
By
2a.
)
1
(
..........
9
4
0
9
4
0
4
2
9
2
1
4
9
:
ation
differenti
implicit
By
1.
.
1
4
9
hyperbola
he
Consider t
ellipse.
the
of
that
similar to
are
methods
This
2
2
2
2
2
2
t
t
dx
dy
t
e
e
dt
dy
t
e
e
dt
dx
e
e
t
y
e
e
t
x
t
t
t
t
t
dx
dy
t
dt
dy
t
t
dt
dx
t
y
t
x
y
x
dx
dy
dx
dy
y
x
dx
dy
y
x
y
x
y
x
x
x
x
x
x
x
x
x
34. The gradient function of the hyperbola: generalised.
)
3
.........(
sinh
cosh
sinh
,
cosh
s
coordinate
parametric
With
)
2
.........(
tan
sec
tan
,
sec
s
coordinate
parametric
With
)
1
(
..........
:
that
see
to
difficult
not
is
it
1
hyperbola
the
general
For the
2
2
2
2
2
2
t
a
t
b
dx
dy
t
b
y
t
a
x
t
a
t
b
dx
dy
t
b
y
t
a
x
y
a
x
b
dx
dy
b
y
a
x
35. The equations of the tangent and normal of the hyperbola.
(2)
into
sub
now
;
2
cosh
;
1
sinh
)
2
,
2
(3
)
sinh
2
,
cosh
3
(
in
find
Or
(1)
into
sub
now
;
)
2
,
2
(3
)
tan
2
,
sec
3
(
in
find
Then
tion
differenta
parametric
using
gradient
the
find
to
required
is
it
If
.
n.b
)
2
.......(
0
cosh
2
6
sinh
6
2
2
3
cosh
2
6
2
2
sinh
6
3
)
cosh
3
(
2
2
)
sinh
2
(
3
)
cosh
3
(
3
2
2
)
sinh
2
(
:
is
tangent
the
of
equation
An
)
sinh
2
,
cosh
3
(
are
s
coordinate
parametric
the
If
3.
)
1
......(
0
sec
2
6
tan
6
2
2
3
sec
2
6
2
2
tan
6
3
)
sec
3
(
2
2
)
tan
2
(
3
)
sec
3
(
3
2
2
)
tan
2
(
:
is
tangent
the
of
equation
An
)
tan
2
,
sec
3
(
are
s
coordinate
parametric
the
If
2.
...(*)
0
6
2
2
3
12
2
2
6
3
)
2
3
(
2
2
)
2
(
3
)
2
3
(
3
2
2
)
2
(
:
is
tangent
the
of
equation
an
So
3
2
2
2
9
2
3
4
9
4
tangent
the
of
gradient
The
:
form
Cartesian
In
1.
:
ways
3
in
determined
be
can
course,
of
),
2
,
2
(3
at
hyperbola
this
o
tangent t
The
.
1
4
9
hyperbola
on the
)
2
,
2
(3
point
he
Consider t
4
2
2
t
t
t
t
t
t
t
t
t
t
t
x
y
t
x
t
y
t
x
t
y
t
x
t
y
t
t
t
t
x
y
t
x
t
y
t
x
t
y
t
x
t
y
t
t
x
y
x
y
x
y
x
y
y
x
dx
dy
y
x
v
v
v
v
v
36. Example:A locus problem involving tangents
)
2
......(
0
cos
sin
0
)
cos
(sin
cos
sin
)
cos
(
sin
cos
)
sin
(
is
of
equation
sin
cos
0
2
2
ation
differenti
implicit
,
circle
the
For
)
1
......(
0
cos
sin
0
)
cos
(sin
cos
sin
)
cos
(
sin
cos
)
sin
(
is
of
equation
an
So
sin
cos
sin
cos
0
2
2
ation
differenti
implicit
1,
ellipse
the
For
.
and
of
equations
parametric
the
find
to
need
we
Here
:
Solution
varies.
as
of
locus
the
Find
.
at
meet
and
).
sin
,
cos
(
point
at the
tangent
has
circle
the
and
)
sin
,
cos
(
point
at the
tangent
has
1
ellipse
The
2
2
2
2
2
2
2
2
1
2
2
2
2
2
2
2
2
2
2
2
1
2
1
2
2
2
2
1
2
2
2
2
a
t
x
t
y
t
t
a
t
x
t
y
t
a
x
t
t
t
a
y
l
t
t
y
x
dx
dy
dx
dy
y
x
a
y
x
ab
t
bx
t
ay
t
t
ab
t
bx
t
ay
t
a
x
t
a
t
b
t
b
y
l
t
a
t
b
t
b
a
t
a
b
y
a
x
b
dx
dy
dx
dy
b
y
a
x
b
y
a
x
l
l
t
P
P
l
l
t
a
t
a
l
a
y
x
t
b
t
a
l
b
y
a
x
Contd >>
37. Example:Alocus problem involving tangents (contd)
0)
(
axis
just the
is
varies
as
of
locus
the
)
0
,
cos
(
s
coordinate
has
0
0
sin
)
(
)
2
......(
0
cos
sin
)
1
......(
0
cos
sin
assume
must
we
ellipse
for the
cos
)
(
cos
)
(
cos
)
(
)
2
......(
0
cos
sin
)
1
......(
0
cos
sin
:
(2)
and
(1)
solve
we
find
To
)
2
......(
0
cos
sin
is
of
equation
The
)
1
......(
0
cos
sin
is
of
equation
The
:
Solution
varies.
as
of
locus
the
Find
.
at
meet
and
).
sin
,
cos
(
point
at the
tangent
has
circle
the
and
)
sin
,
cos
(
point
at the
tangent
has
1
ellipse
The
2
2
2
1
2
1
2
2
2
2
1
2
2
2
2
y
x
t
P
t
a
P
y
t
y
b
a
b
ab
t
bx
t
by
ab
t
bx
t
ay
b
a
t
a
x
a
b
a
t
x
a
b
a
ab
t
x
a
b
a
a
t
ax
t
ay
ab
t
bx
t
ay
P
a
t
x
t
y
l
ab
t
bx
t
ay
l
t
P
P
l
l
t
a
t
a
l
a
y
x
t
b
t
a
l
b
y
a
x
38. Exercises: Tangents and normals of the ellipse and hyperbola.
)
tan
3
,
sec
5
(
at
1
9
25
2.
)
sin
,
cos
2
(
at
1
4
1.
:
specified
point
at the
sections
conic
following
the
of
normal
and
tangent
the
of
equations
the
Find
2
2
2
2
y
x
y
x
39. Exercises: Tangents and normals of the ellipse and hyperbola
(solutions).
0
cos
sin
3
sin
2
cos
cos
sin
4
sin
2
cos
sin
cos
)
cos
2
(
sin
2
)
sin
(
cos
)
cos
2
(
cos
sin
2
)
sin
(
:
is
normal
the
of
equation
The
0
2
cos
sin
2
0
)
cos
(sin
2
cos
sin
2
cos
2
cos
sin
2
sin
2
)
cos
2
(
cos
)
sin
(
sin
2
)
cos
2
(
sin
2
cos
)
sin
(
:
is
tangent
the
of
equation
an
So
sin
2
cos
sin
4
cos
2
),
sin
,
cos
2
(
At
4
tangent
the
of
gradient
The
0
2
4
2
1
1
4
.
1
2
2
2
2
2
2
x
y
x
y
x
y
x
y
x
y
x
y
x
y
x
y
x
y
dx
dy
y
x
dx
dy
dx
dy
y
x
y
x
40. Exercises: Tangents and normals of the ellipse and hyperbola
(solutions).
0
tan
sec
34
tan
5
sec
3
tan
sec
25
tan
5
tan
sec
9
sec
3
)
sec
5
(
tan
5
)
tan
3
(
sec
3
)
sec
5
(
sec
3
tan
5
)
tan
3
(
:
is
normal
the
of
equation
An
0
15
sec
3
tan
5
0
)
tan
(sec
15
sec
3
tan
5
sec
15
sec
3
tan
15
tan
5
)
sec
5
(
sec
3
)
tan
3
(
tan
5
)
sec
5
(
tan
5
sec
3
)
tan
3
(
:
is
tangent
the
of
equation
an
So
tan
5
sec
3
tan
3
25
sec
5
9
),
tan
3
,
sec
5
(
At
25
9
tangent
the
of
gradient
The
0
9
2
25
2
1
9
25
.
2
2
2
2
2
2
2
x
y
x
y
x
y
x
y
x
y
x
y
x
y
x
y
x
y
dx
dy
y
x
dx
dy
dx
dy
y
x
y
x