1.) Write the equation of a circle in standard form with radius 3 and center (-3,4). Then graph the circle.

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1 Welcome to the world of conic sections! Some examples of conics in the real world: Parabolas Ellipse Hyperbola Your Assignment: Circle -Find at least four pictures (different from the ones above) of conic sections in the real world. You must have at least one picture of each type of conic section. -Identify what each pictures represents - parabola, hyperbola, circle or ellipse. -Glue to 8 x 10 in. or 9 x 12 in. paper. -Write your name on the front or back. -Be creative! -Due Tuesday, April 17th Apr 4 11:19 AM Chapter 10: Conic Sections Circles Standard Form of the Equation of a Circle: (x - h) 2 + (y - k) 2 = r 2 r is the and (h,k) is the. Examples: 1.) Write the equation of a circle in standard form with radius 3 and center (-3,4). Then graph the circle. Apr 4 10:43 AM 1

2 2.) Write the equation of the circle in standard form whose center is (6,1) and is tangent to the y-axis. Then graph the circle. General Form of the Equation of a Circle: x 2 + y 2 + Dx + Ey + F = 0 D, E and F are. Midpoint of a Line Segment: If the coordinates of P 1 and P 2 are (x 1,y 1 ) and (x 2,y 2 ), respectively, then the midpoint of P 1 P 2 has coordinates (x 1 + x 2, y 1 + y 2 ). 2 2 Apr 4 10:57 AM Review: Completing the square 1.) x 2 + 4x - 5 = 0 2.) 2x 2 + 4x - 8 = 0 Examples: Write the equation of the circle in standard form. Then graph each circle. 1.) x 2 + y 2-4x + 12y + 30 = 0 Apr 4 10:59 AM 2

3 2.) 2x 2 + 2y 2-4x + 12y - 18 = 0 Distance Formula for Two Points: The distance between two points (x 1,y 1 ) and (x 2,y 2 ) in the coordinate plane is given by d = (x 2 - x 1 ) 2 + (y 2 - y 1 ) 2. Example: Find the distance between the points (-1,-6) and (5,-3). Apr 4 11:03 AM Examples: Write the equation of the circle that satisfies each set of conditions. 1.) The circle passes through the point (5,6) and has its center at (-4,3). 2.) The endpoints of a diameter are at (2,3) and at (-6,-5). Apr 4 11:16 AM 3

4 Ellipses B -Ellipse: D C F 1 F 2 A -Foci: E -Center: -Minor Axis: -Major Axis: -Vertices: Apr 4 11:18 AM Standard Form of the Equation of Orientation Description and Ellipse (x - h) 2 + (y - k) 2 = 1, -Center: (h,k) a 2 b 2 - Foci: (h±c, k) where c 2 = a 2 - b 2. -Major Axis: y = k (h,k) -Major Axis y = k Vertices: (h±a, k) -Minor Axis: x = h -Minor Axis x = h Vertices: (h, k±b) (y - k) 2 + (x - h) 2 = 1, -Center: (h,k) a 2 b 2 -Foci: (h, k±c) where c 2 = a 2 - b 2. -Major Axis: x = h (h,k) y = k -Major Axis Vertices: (h, k±a) -Minor Axis: y = k -Major Axis Vertices: (h±b, k) x = h Apr 4 11:56 AM 4

5 Examples: 1.) Consider the ellipse graphed at the right. (2, 7) (2, 4) (8, 4) a.) Write the equation of the ellipse in standard form. b.) Find the coordinates of the foci. Apr 5 9:09 AM 2.) For the equation (y - 3) 2 + (x + 4) 2 = 1, find the coordinates 25 9 of the center, foci and vertices of the ellipse. Then graph. Apr 5 9:15 AM 5

6 3.) Find the coordinates of the center, the foci and the vertices of the ellipse with the equation 4x 2 + 9y 2-40x + 36y = 0. Then graph the ellipse. Apr 5 9:17 AM Hyperbolas asymptote center asymptote -Hyperbola: transverse axis F2 F1 vertices -Foci: conjugate axis -Center: -Vertex: -Asymptotes: -Transverse Axis: -Conjugate Axis: Apr 5 9:21 AM 6

7 Standard Form of the Equation of a Orientation Description Hyperbola (x - h) 2 - (y - k) 2 = 1, Center: (h,k) a 2 b 2 Foci: (h±c, k) where b 2 = c 2 - a 2. y = k (h,k) Vertices: (h±a, k) Equation of transverse axis: y = k (parallel to x-axis) x = h Asymptotes: y - k = ±(b/a)(x-h) (y - k) 2 - (x - h) 2 = 1, Center: (h, k) a 2 b 2 Foci: (h, k±c) where b 2 = c 2 - a 2. Vertices: (h, k±a) (h,k) Equation of transverse y = k axis: x = h (parallel to y-axis) Asymptotes: y - k = ±(a/b)(x-h) x = h Apr 5 10:22 AM Examples: 1.) Find the coordinates of the center, the foci, the vertices and the equations of the asymptotes of the hyperbola whose equation is x 2 - y 2 = 1. Then graph Apr 5 10:45 AM 7

8 2.) Find the coordinates of the center, foci, vertices and the equations of the asymptotes of the graph of 9x 2-4y 2-54x - 40y - 55 = 0. Then graph. Apr 5 11:17 AM Parabolas vertex -Parabola: axis of symmetry focus -Focus: directrix -Directrix: -Axis of symmetry: -Vertex: Apr 5 11:21 AM 8

9 Standard Form of the Equation of a Parabola Orientation Description (y - k) 2 = 4p(x - h) (h, k) Vertex: (h, k) y = k (h + p, k) Focus: (h + p, k) Axis of symmetry: y = k Directrix: x = h - p Opening: Right if p > 0 x = h - p Left if p < 0 (x - h) 2 = 4p(y - k) Vertex: (h, k) Focus: (h, k + p) Axis of symmetry: x = h Directrix: y = k - p Opening: Upward if p > 0 (h, k) Downward if p < 0 (h, k + p) x = h y = k - p Apr 5 11:45 AM Examples: For the equation of each parabola, find the coordinates of the vertex and focus and the equations of the directrix and axis of symmetry. Then graph. 1.) x 2 = 12(y - 1) Apr 5 12:02 PM 9

10 2.) y 2-4x + 2y + 5 = 0 Apr 5 12:29 PM Examples: Write the equation of the parabola that meets each set of conditions. Then graph. 1.) The vertex is at (-5,1) and the focus is at (2,1). 2.) The axis of symmetry is y = 6, the focus is at (0,6) and p = -3. Apr 5 12:31 PM 10

11 Chapter 10 Homework Name: 1.) Write the standard form of the equation of the circle with center (4,-7) and radius 3. Then graph. 2.) Write the standard form of the equation of the circle graphed below. Apr 5 12:37 PM 3.) Write the standard form of the equation of the circle. Then graph. 6x 2-12x + 6y y = 36 4.) Write the equation of the circle whose endpoints of a diameter are (-3,4) and (2,1). Apr 5 12:48 PM 11

12 For each equation of the ellipse, find the coordinates of the center, foci and vertices. Then graph each equation. 5.) x 2 + (y - 4) 2 = ) 9x 2 + 4y 2-18x + 16y = 11 Apr 5 12:57 PM Write the equation of each ellipse in standard form. Then find the coordinates of the foci. 7.) 8.) Apr 5 1:01 PM 12

13 9.) Determine which of the following equations matches the graph of the hyperbola below. A.) x 2 - y 2 = 1 4 B.) y 2 - x 2 = 1 4 C.) x 2 - y 2 = 1 4 D.) y 2 - x 2 = ) Write the equation of a hyperbola centered at the origin, with a = 8, b = 5 and transverse axis on the y-axis. Apr 5 1:08 PM For the equation the hyperbola, find the coordinates of the center, the foci and the vertices and the equations of the asymptotes. Then graph. 11.) (y - 3) 2 - (x - 2) 2 = Apr 5 1:27 PM 13

14 For the equation of each parabola, find the coordinates of the vertex and focus, and the equations of the directrix and axis of symmetry. Then graph the equation. 12.) x 2 + 8x + 4y + 8 = 0 Apr 5 1:32 PM 13.) (y - 6) 2 = 4x 14.) Explain a way in which you might distinguish the equation of a parabola from the equation of a hyperbola. Apr 5 1:36 PM 14