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Section 8.3Ellipses

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 EllipseMajor axis Parallel to x - axis Center = (0, 0)Vertices (a, 0), (- a, 0)Foci (c, 0), (- c, 0)c2 = a2 - b2 Major Axis = 2aMinor Axis = 2b(0, 0)Minor Intercepts (0, b), (0, -b)a > b > 0

Standard Equations for an EllipseMajor axis parallel to y - axis Center = (0, 0)Vertices (0, a), (0, - a)Foci (0, c), (0, - c)Major Axis = 2aMinor Axis = 2bMinor Intercepts (b, 0), (- b, 0)(0, 0)c2 = a2 - b2 a > b > 0

a2 = 16 a = 4b2 = 9 b = 3Minor intercepts = (0, 3) & (0,- 3)Maj. Axis = 2a = 2(4) = 8 Min. Axis = 2b = 2(3) = 6Vertices = (4, 0) & (- 4, 0) Foci = (7, 0) & (- 7, 0)

a2 = 81 a = 9b2 = 16 b = 4Vertices = (0, 9) & (0, - 9) Minor intercepts = (4,0) & (- 4,0)Maj. Axis = 2a = 2(9) = 18 Min. Axis = 2b = 2(4) = 8Foci = (0, 65) & (0, - 65)

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

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

Ellipse1. When Major axis is on x-axis Major axis length = 32 Minor axis length = 30 Therefore, a = 32 2 = 16a2 = 256b = 30 2 = 15 b2 = 225

2. Major axis on y-axis Major axis length = 16 Distance from Foci to Center = 7 EllipseTherefore, c = 7a = 16 2 = 8 a2 = 64c2 = a2 b2 b2 = a2 c2 = 64 49 = 15

Find the equation of the ellipse in the form below

if thee center is the origin.a2 = 100b2 = 36

Translations Ellipses translate just like circles and parabolas doby using h and k in the standard equation.This is for a horizontal major axis, switch a and b for a vertical major axisif your equation isnt in this form you will need to complete the square to make it so

Graph the ellipseCenter: (-1,3)Major axis parallel to x-axisPlace a point 3 units right and left of centerPlace 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-axisCenter is (-4,1)Place 2 points 1.4 unit right and left of centerPlace 2 points 2.8 units up and down from center

Write the equation of the ellipseFoci: (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 = 1Plug into standard form:

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