Lesson 9.2 Ellipses

Ellipse Set of all points where the sum of the distances to two fixed points (foci) is constant. 8cm 7cm 5cm 4cm 9cm 3cm 10cm 2cm Focus Focus

Other parts of an ellipse: Major axis Center Minor axis Vertices Major & Minor axis can run in opposite direction:

a → distance from center to vertex on major axis Deriving the Equation for an Ellipse Equation comes from distances: b a c a → distance from center to vertex on major axis b → distance from center to vertex on minor axis c → distance from center to a focus

Equation of an ellipse derives from distances: Relation among a, b, & c: Equation of an ellipse derives from distances: or

USING THE EQUATION 1) Center is at h and k 2) a is the larger denominator/ b is smaller (major/minor) Use a, b, and/or c to find missing values using the equations Use a, b, and c distances to find foci and vertices from center If a is under x, the ellipse “goes” Left/Right If a is under y, the ellipse “goes” Up/Down

Example Name the foci, center, length of major and minor axes. Then sketch the ellipse.

e close to 1 → more elliptical Eccentricity where 0 < e < 1 small e → circular e close to 1 → more elliptical Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune orbital eccentricity 0.2056 0.0068 0.0167 0.0934 0.0483 0.0560 0.0461 0.0097

Example Find the center, foci, vertices and eccentricity. 9x2 + 4y2 + 36x – 24y + 36 = 0

Example Write the equation for the ellipse with foci (0,0), (0,8); and major axis length of 16.