Section 8.3 Ellipses Parabola Hyperbola Circle Ellipse

Ellipse:

Besides having the two foci, an ellipse also has a major and minor axis, vertices at the end of the major axis and center point where the two axes cross.

Standard Equations for an Ellipse Major axis Parallel to x - axis x 2 y 2 a 2 b 2 + = 1 Center = (0, 0) Vertices (a, 0), (- a, 0) a (a,0) V (- a, 0) V F (- c, 0) F (c, 0) Foci (c, 0), (- c, 0) c 2 = a 2 - b 2 Major Axis = 2a Minor Axis = 2b (0, 0) Minor Intercepts (0, b), (0, -b) b (0, b) (0, - b) a > b > 0

Standard Equations for an Ellipse Major axis parallel to y - axis x 2 y 2 b 2 a 2 + = 1 Center = (0, 0) Vertices (0, a), (0, - a) Foci (0, c), (0, - c) Major Axis = 2a Minor Axis = 2b Minor Intercepts (b, 0), (- b, 0) a (0,a) V (0,- a,) V b (-b,0) (b,0) (0, 0) (0,c) F F (0,-c) c 2 = a 2 - b 2 a > b > 0

a 2 = 16 a = 4 b 2 = 9 b = 3 c 2 = a 2 - b 2 = = 7 c =  7 Minor intercepts = (0, 3) & (0,- 3) Maj. Axis = 2·a = 2(4) = 8 Min. Axis = 2·b = 2(3) = 6 | | | | | | | || | | || | | | | | | | | | | Ellipse Sketch, Find Foci, Length of Minor and Major Axis For Center at the origin. x 2 y =  7 7 7 Vertices = (4, 0) & (- 4, 0) Foci = (  7, 0) & (-  7, 0)

a 2 = 81 a = 9 b 2 = 16 b = 4 c 2 = a 2 - b 2 = = 65 c =  65 Vertices = (0, 9) & (0, - 9) Minor intercepts = (4,0) & (- 4,0) Maj. Axis = 2·a = 2(9) = 18 Min. Axis = 2·b = 2(4) = 8 | | | | | | | || | | || | | | | | | | | | | Ellipse Sketch, Find Foci, Length of Minor and Major Axis For Center at the origin. x 2 y =  65  65 Foci = (0,  65) & (0, -  65)

Graph the Ellipse Needs to be set equal to 1. Vertices: (0,-4) and (0,4) Minor Intercepts: (-1,0) and (1,0)

Find the equation of the ellipse Foci: (-1,0) and (1,0) Vertices: (-3,0) and (3,0) Therefore a = 3and c = 1

Ellipse Find an equation of an ellipse in the form x 2 y 2 a 2 b 2 + = 1 1. When Major axis is on x-axis Major axis length = 32 Minor axis length = 30 Therefore, a = 32 ÷ 2 = 16 a 2 = 256 b = 30 ÷ 2 = 15 b 2 = 225 x 2 y = 1

2. Major axis on y-axis Major axis length = 16 Distance from Foci to Center = 7 Ellipse Therefore, c = 7 Find an equation of an ellipse in the form x 2 y 2 b 2 a 2 + = 1 a = 16 ÷ 2 = 8 a 2 = 64 c 2 = a 2 – b 2 b 2 = a 2 – c 2 = 64 – 49 = 15 x 2 y = 1

| | | | | | | || | | || | | | | | | | | | | Find the equation of the ellipse in the form below if thee center is the origin. x 2 y 2 a 2 b 2 + = 1 a = 10 b = 6 a 2 = 100 b 2 = 36 x 2 y = 1

Translations Ellipses translate just like circles and parabolas do…by using h and k in the standard equation. This is for a horizontal major axis, switch a and b for a vertical major axis…if your equation isn’t in this form you will need to complete the square to make it so…

Graph the ellipse Center: (-1,3) Major axis parallel to x-axis Place a point 3 units right and left of center Place a point 1 unit above and below the center. Foci are about 2.8 units to the left and right of center.

Graph the ellipse

Major axis is parallel to the y-axis Center is (-4,1) Place 2 points 1.4 unit right and left of center Place 2 points 2.8 units up and down from center

Write the equation of the ellipse Foci: (2,-2) and (4,-2) Vertices: (0,-2) and (6,-2) Center is halfway between the vertices so the point (3,-2) We know a = 3 and c = 1 Plug into standard form:

Write the equation of the ellipse Major axis vertical with length of 6 and minor axis length of 4 centered at (1,-4)