2.) Write the standard form of the equation of a circle whose endpoints of diameter are (4, 7) and (2,3).
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1 Ch 10: Conic Sections Name: Objectives: Students will be able to: -graph parabolas, hyperbolas and ellipses and answer characteristic questions about these graphs. -write equations of conic sections Dec 1 6:05 PM Circles Standard Form: 1.) Write the equation of a circle with radius 5 and center (-3,2). 2.) Write the standard form of the equation of a circle whose endpoints of diameter are (4, 7) and (2,3). Nov 23 6:43 PM 1
2 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 a is bigger than b. 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) -Major Axis y = k a is bigger than b. Vertices: (h,k±a) -Minor Axis: y = k -Minor Axis Vertices: (h±b,k) x = h Apr 4 11:56 AM 2
3 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 3
4 3.) Find the coordinates of the center, foci and vertices of the ellipse with the equation 4x 2 + 9y 2-40x + 36y = 0. Then graph the ellipse. Apr 5 9:17 AM 4.) Find the equation in standard form of the ellipse whose endpoints of axes are (±7,0) and (0,±4). Feb 18 2:42 PM 4
5 Hyperbolas asymptote center asymptote -Hyperbola: transverse axis F 2 F 1 vertices -Foci: conjugate axis -Center: -Vertex: -Asymptotes: -Transverse Axis: -Conjugate Axis: Apr 5 9:21 AM 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) y = k (h,k) -Vertices: (h±a,k) a is not necessarily -Equation of transverse bigger than b. a comes first. axis: y = k 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) -Vertices: (h,k±a) a is not necessarily (h,k) y = k -Equation of transverse bigger than b. axis: x = h comes first. -Asymptotes: y - k = ±(a/b)(x - h) x = h Apr 5 10:22 AM 5
6 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 2.) Find the coordinates of the center, foci, vertices, eccentricity 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 6
7 4.) Write the equation in standard form for the hyperbola whose transverse axis endpoints are (5,3) and (-7,3) and conjugate axis is length 10. Feb 18 2:43 PM Parabolas vertex -Parabola: axis of symmetry focus -Focus: directrix -Directrix: -Axis of symmetry: -Vertex: Apr 5 11:21 AM 7
8 Standard Form of the Equation of a Parabola Orientation Description (y - k) 2 = 4p(x - h) y = k (h, k) x = h - p (h + p, k) Vertex: (h,k) Focus: (h + p, k) Axis of symmetry: y = k Directrix: x = h - p Opening: Right if p > 0 Left if p < 0 (x - h) 2 = 4p(y - k) Vertex: (h,k) (h, k + p) (h, k) y = k - p x = h Focus: (h, k + p) Axis of symmetry: x = h Directrix: y = k - p Opening: Up if p > 0 Down if p < 0 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 8
9 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 9
10 HW solutions are posted online. Nov 23 6:51 PM Nov 23 6:51 PM 10
11 Ch 10 Homework Name: For each equation of the ellipse, find the coordinates of the center, foci and vertices. Then graph each equation. 1.) 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. 3.) 4.) Apr 5 1:01 PM 11
12 5.) Write the equation of the hyperbola below. 6.) 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, vertices and the equations of the asymptotes. Then graph. 7.) (y - 3) 2 - (x - 2) 2 = Apr 5 1:27 PM 12
13 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. 8.) x 2 + 8x + 4y + 8 = 0 Apr 5 1:32 PM 9.) (y - 6) 2 = 4x 10.) Explain a way in which you might distinguish the equation of a parabola from the equation of a hyperbola. Apr 5 1:36 PM 13
14 Write the equation in standard form for the conic with the given characteristics. 11.) Parabola: Focus: (0,5), Directrix: y = ) Parabola: Focus (-2,-4), Vertex: (-4,-4) 13.) Ellipse: Major axis length 6 on y-axis, minor axis length 4 Feb 18 2:55 PM 14.) Ellipse: Minor axis endpoints: (0,±4), Major axis length ) Hyperbola: Foci (±5, 0), Transverse axis length 3 16.) Hyperbola: Transverse axis endpoints (-1,3) and (5,3), slope of one asymptote is 4/3. Feb 18 2:58 PM 14
15 16.) Write the standard form of the equation of a circle with radius 5 and center (-2,3). Then graph. 17.) Write the standard form of the equation of a circle whose endpoints of diameter are (-2, 3) and (6, 5). Nov 23 6:49 PM 18.) Write and solve your own conic section problem. :) Nov 23 6:51 PM 15
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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