Conic Sections and Analytic Geometry

Transcription

1 Chapter 9 Conic Sections and Analytic Geometry

2 Chapter 9 Conic Sections and Analytic Geometry 9.1 The Ellipse 9.2 The Hyperbola 9.3 The Parabola 9.4 Rotation of Axes 9.5 Parametric Equations 9.6 Conic Sections in Polar Coordinates

3 Chapter 9 Conic Sections and Analytic Geometry 9.1 The Ellipse 9.2 The Hyperbola 9.3 The Parabola 9.4 Rotation of Axes 9.5 Parametric Equations 9.6 Conic Sections in Polar Coordinates

4 Chapter 9. Conic Sections and Analytic Geometry Geometric Definitions of Conic Sections and Their Standard Equations Each conic secteach conic section (or simply conic) can be described as the intersection of a plane and a double-napped cone. There are seven different possible intersections. Three Others When the intersecting plane passes through the vertex of the cone. Four Basic Conics For four basic conics, the intersecting plane does not pass through the vertex of the cone.

5 Chapter 9. Conic Sections and Analytic Geometry Geometric Definitions of Conic Sections and Their Standard Equations Parabolas, ellipses, and hyperbolas General second-degree equation Ax 2 + Bxy + Cy 2 + Dx + Ey + F = 0 1. The paths of the planets around the sun (ellipses) 2. Parabolic mirrors are used to converge light beams at the focus of the parabola 3. Parabolic microphones perform a similar function with sound waves 4. Parabola is used in the design of car headlights and in spotlights 5. Trajectory of objects thrown or shot near the earth s surface will follow a parabolic path 6. Hyperbolas are used in a navigation system known as LORAN(long range navigation) 7. Hyperbolic as well as parabolic mirrors and lenses are used in systems of telescopes.

6 Chapter 9. Conic Sections and Analytic Geometry In this section, we study their exact mathematical definition in which each of the conics is defined as a locus (collection) of points satisfying a certain geometric property. For example, a circle can be defined as the collection of all points (x, y) that are equidistant from a fixed point (h, k). This locus definition easily produces the standard equation of a circle x h 2 + y k 2 = r 2.

7 9.1 The Ellipse Objectives: Graph ellipses centered at the origin. Write equations of ellipses in standard form. Graph ellipses not centered at the origin. Solve applied problems involving ellipses.

8 9.1 The Ellipse Ellipse An ellipse is the set of points in a plane the sum of whose distances from two fixed points F 1 and F 2 is a constant. These two fixed points are called the foci (plural of focus). The midpoint of the line segment connecting the foci is called the center. The line passing through two foci is called the major axis. The perpendicular line to the major axis passing through the midpoint of two foci is called the minor axis. The two intersection points of the major axis and the ellipse are called vertices. minor axis vertex ( a, 0) (0, b) P(x, y) (0,0) F 1 F 2 ( c, 0) (c, 0) (0, b) vertex (a, 0) major axis Standard Equation Let the sum of the distances from a point on the ellipse to the foci be 2a. Then x 2 a 2 + y2 b 2 = 1 where c 2 = a 2 b 2 < a 2.

9 9.1 The Ellipse vertex ( a, 0) ( b, 0) (0, b) minor axis (0,0) P(x, y) F 1 F 2 ( c, 0) (c, 0) (0, b) vertex (0, a) (0, c) F 1 F (0, c) 2 major axis vertex (0,0) (0, a) vertex (a, 0) P(x, y) (b, 0) major axis minor axis Standard Equation Let the sum of the distances from a point on the ellipse to the foci be 2a. Then x 2 a 2 + y2 b 2 = 1 where b 2 = a 2 c 2 < a 2. Standard Equation Let the sum of the distances from a point on the ellipse to the foci be 2a. Then x 2 b 2 + y2 a 2 = 1 where b 2 = a 2 c 2 < a 2.

10 9.1 The Ellipse Standard Equation of an Ellipse with Translations The standard form of the equation of a parabola with center (h, k) and major and minor axes of lengths 2a and 2b, a > b, is x h 2 y k 2 a 2 + b 2 = 1 Major axis is horizontal or x h 2 2 y k b 2 + a 2 = 1 Major axis is vertical The foci lie on the major axis, c units from the center, with c 2 = a 2 b 2 The ration e = c/a is called the eccentricity of an ellipse.

11 9.1 The Ellipse Example 1. Graph and locate the foci: x2 9 + y2 4 = 1.

12 9.1 The Ellipse Example 2. Graph and locate the foci: 25x y 2 = 400.

13 9.1 The Ellipse Example 3. Find the strandard form of the equation of an ellipse with foci at ( 1, 0) and (1, 0) and vertices 2, 0 and 2, 0.

14 9.1 The Ellipse Example 4. Graph x y = 1. Where are the foci located?

15 9.1 The Ellipse Example 5. (Graphing an Ellipse Centered at (h, k) by Completing the Square Graph 4x y x 288y = 0. Where are the foci located?

16 9.1 The Ellipse Applications of Ellipse Movements of Planets in Our Solar System The planets in our solar system move in elliptical orbits, with the Sun at a focus. Earth satellites also travel in elliptic orbits, with Earth at a focus.

17 9.1 The Ellipse Applications of Ellipse Reflection Property of an Ellipse Let P be a point on an ellipse. The tangent line to the ellipse at point P makes equal angles with the lines through P and the foci.

18 9.1 The Ellipse Applications of Ellipse Constructive Design Example 6. A semielliptical archway over a one-way road has a height of 10 eet and a width of 40 feet. Your truck has a width of 10 feet and a height of 9 feet. Will your truck clear the opening of the archway? End of 9.1

19 9.2 The Hyperbola Objectives: Locate a hyperbola s vertices and foci. Write equations of hyperbolas in standard form. Graph hyperbolas centered at the origin. Graph hyperbolas not centered at the origin. Solve applied problems involving hyperbolas.

20 9.2 The Hyperbola Hyperbolas An hyperbola is the set of points in a plane the difference of whose distances from two fixed points F 1 and F 2 is a constant. These two fixed points are called the foci. The peaks are called the vertices of the hyperbola. An hyperbola has two slant asymptotes: y = b a x and y = b a x y = b a x P(x, y) (0,0) ( c, 0) (c, 0) F 1 F 2 vertices ( a, 0) and (a, 0) y = b a x Standard Equation Let the difference of the distances from a point on the ellipse to the foci be ±2a. Then x 2 a 2 y2 b 2 = 1 where c 2 = a 2 + b 2.

21 9.2 The Hyperbola P(x, y) (0,0) ( c, 0) (c, 0) F 1 F 2 vertices ( a, 0) and (a, 0) y = b a x y = b a x Standard Equation Let the difference of the distances from a point on the ellipse to the foci be ±2a. Then x 2 a 2 y2 b 2 = 1 where c 2 = a 2 + b 2. Asymptotes: y = ± b/a x. P(x, y) vertices (0, a) and (0, a) (0, c) F 2 (0,0) F 1 (0, c) y = a b x y = a b x Standard Equation Let the difference of the distances from a point on the ellipse to the foci be ±2a. Then y 2 a 2 x2 b 2 = 1 where c 2 = a 2 + b 2. Asymptotes: y = ± a/b x.

22 9.2 The Hyperbola Standard Equation of a Hyperbola The standard form of the equation of a parabola with center at (h, k) is x h 2 2 y k a 2 b 2 = 1 Transverse axis is horizontal or y k 2 a 2 x h 2 b 2 = 1 Transverse axis is vertical The vertices are a units from the center, and the foci are c units from the center, where c 2 = a 2 + b 2. The eccentricity e of a hyperbola is given by the ratio e = c/a (> 1). Asymptotes of a Hyperbola For a horizontal transverse axis, the equation of the asymptotes are y = k + b a x h and y = k b a x h. For a vertical transverse axis, the equation of the asymptotes are y = k + a b x h and y = k a b x h.

23 9.2 The Hyperbola Example 1. Find the vertices and locate the foci for each of the following hyperbolas with the given equation: a x2 16 y2 9 = 1 b y2 9 x2 16 = 1

24 9.2 The Hyperbola Example 2. Find the standard form of the equation of a hyperbola with foci at (0, 3) and (0, 3) and vertices (0, 2) and (0, 2).

25 9.2 The Hyperbola Graphing Hyperbola 1. Locate the vertices Use DASHED lines to draw the rectangle centered at the origin with sides parallel to the axes, crossing one axis at ±a and the other at ±b. 3. Use dashed lines to draw the diagonals of this rectangle and extend them to obtain the asymptotes. 4. Draw the two branches of the hyperbola by starting at each vertex and approaching the asymptotes. Example 3. Graph and locate the foci: x2 y asymptotes. = 1. What are the equations of the

26 9.2 The Hyperbola Example 4. Graph and locate the foci: 9y 2 4x 2 = 36. What are the equations of the asymptotes?

27 9.2 The Hyperbola Example 5. Graph x y equations of the asymptotes? = 1. Where are the foci located? What are the

28 9.2 The Hyperbola Example 6. Graph 4x 2 24x 25y y 489 = 0. Where are the foci located? What are the equations of the asymptotes?

29 9.2 The Hyperbola Applications of the Hyperbola When a jet flies at a speed greater than the speed of sound, the shock wave that is created is heard as a sonic boom. The wave has the shape of a cone. The shape formed as the cone hits the ground is one branch of a hyperbola. Halley s Comet, a permanent part of our solar system, travels around the Sun in an elliptic orbit. Other comets pass through the solar system only once, following a hyperbolic path with the Sun as a focus.

30 9.2 The Hyperbola Example 7. (Application) An explosion is recorded by two microphones that are 2 miles apart. Microphone M 1 received the sound 4 seconds before microphone M 2. Assuming sound travels at 1100 feet per second, determine the possible locations of the explosion relative to the location of the microphones. End of 9.2

31 9.3 The Parabolas Objectives: Graph parabolas with vertices at the origin. Write equations of parabolas in standard form. Graph parabolas with vertices not at the origin. Solve applied problems involving parabolas. Identity conics without completing the square.

32 9.3 The Parabolas What we have learned about a parabola in Algebra class: y = a x h 2 + k or y = a x h 2 + k Note h = b b and k = f. 2a 2a Graphing Parabolas Graphing y = a x h 2 + k or y = a x h 2 + k If a > 0, the graph opens upward. If a < 0, the graph opens downward. The vertex of y = a x h 2 + k is (h, k). The x-coordinate of the vertex of y = ax 2 + bx + c is x = b 2a

33 9.3 The Parabolas Definition of Parabolas A parabola is a set of points in a plane that are equidistance from a fixed point F (called the focus) and a fixed line (called the directrix). The point halfway between the focus and the directrix lies on the parabola and it is called the vertex. The line through the focus perpendicular to the directrix is called the axis of the parabola. (0, p) F (focus) (0,0) vertex axis (x, y) directrix y = p Standard Equation An equation of the parabola with focus (0, p) and directrix y = p is x 2 = 4py (See Figure)

34 9.3 The Parabolas (0, p) F (focus) (0,0) vertex axis (x, y) directrix y = p Standard Equation An equation of the parabola with focus (0, p) and directrix y = p is x 2 = 4py (See Figure) vertex axis (0,0) (p, 0) F (focus) Standard Equation An equation of the parabola with focus (p, 0) and directrix x = p is y 2 = 4px directrix x = p (x, y)

35 9.3 The Parabolas Standard Equation of a Parabola The standard form of the equation of a parabola with vertex (h, k) and directrix y = k p is x h 2 = 4p y k Vertical axis For directrix x = h p, the equation is y k 2 = 4p x h Horizontal axis The focus lies on the axis p units (directed distance) from the vertex. The coordinates of the focus are as follows h, k + p Vertical axis h + p, k Horizontal axis

36 9.3 The Parabolas Example 1. Find the focus and directrix of the parabola given by y 2 = 12x. Then graph the parabola.

37 9.3 The Parabolas The Latus Rectum and Graphing Parabolas The latus rectum of a parabola is a line segment that passes through its focus, is parallel to its directrix, and has its endpoints on the parabola. The length of the latus rectum for the graphs of y 2 = 4px and x 2 = 4py is 4p. Example 2. Find the focus and directrix of the parabola given by x 2 = 8y. Then graph the parabola. Include the latus rectum on the graph.

38 9.3 The Parabolas Example 3. Find the standard form of the equation of a parabola with focus (5, 0) and directrix x = 5.

39 9.3 The Parabolas Example 4. Find the vertex, focus, and directrix of the parabola given by Then graph the parabola. x 3 2 = 8 y + 1.

40 9.3 The Parabolas Example 5. Find the vertex, focus, and directrix of the parabola given by Then graph the parabola. y 2 + 2y + 12x 23 = 0.

41 9.3 The Parabolas Applications of the Hyperbola Cables hung between structures to form suspension bridges form parabolas. Arches constructed of steel and concrete, whose main purpose is strength, are usually parabolic in shape Some comets follow parabolic paths. Only comets with elliptical orbits, such as Halley s Comet, return to our part of the galaxy Light originates at the focus. The refective properties of parabolic surfaces are used in the design of searchlights and automobile headlights.

42 9.3 The Parabolas Example 6. An engineer is designing a flashlight using a parabolic reflecting mirror and a light source. The casting has a diameter of 4 inches and a depth of 2 inches. What is the equation of the parabola used to shape the mirror? At what point should the light source be placed relative to the mirror s vertex?

43 Geometric Definitions of Conic Sections and Their Standard Equations Parabolas, ellipses, and hyperbolas General second-degree equation Ax 2 + Bxy + Cy 2 + Dx + Ey + F = 0

44 Identifying a Conic Section without Completing the Square A nondegenerate conic section of the form Ax 2 + Cy 2 + Dx + Ey + F = 0 in which A and C are not both zero, is a circle if A = C, a parabola if AC = 0. an ellipse if A C and AC > 0, and a hyperbola if AC < 0. Example 6. Identify the graph of each of the following nondegenerate conic sections: (a) 4x 2 25y 2 24x + 250y 489 = 0 (b) x 2 + y 2 + 6x 2y + 6 = 0 (c) y x + 2y 23 = 0 (d) 9x y 2 54x + 50y 119 = 0 End of 9.3