10.3 Ellipses JMerrill, 2010. General Second Degree Equation Ax 2 + Bxy + Cy 2 + Dx + Ey + F = 0.

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Presentation transcript:

10.3 Ellipses JMerrill, 2010

General Second Degree Equation Ax 2 + Bxy + Cy 2 + Dx + Ey + F = 0

Definition - Ellipse An ellipse is the set of all points in a plane, the sum of whose distances from two distinct fixed points (foci) is constant.

Standard Equations of Ellipses

Writing Equations from Descriptions Write an equation of the ellipse whose vertices are at (-3, 0) and (3, 0) and whose minor axis length is 4. Find the foci. You Try: Write an equation of the ellipse whose vertices are (-5, 0), (5, 0) and whose co- vertices are (0, -3), (0, 3).Find the foci.

Writing Equations from Graphs You Try:

Writing the Equation in Standard Form & Graph Sketch the ellipse given by x 2 + 4y 2 + 6x – 8y + 9 = 0 In order to put this in standard form, you must complete the square: 1.Move the 9 to the right 2.Group the x’s. Group the y’s. 3.You MUST have a leading coefficient of 1. If it’s not 1, factor out the coefficient. 4.Complete the square 5.The equation must = 1; divide

You Try Put the ellipse 4x 2 + y 2 - 8x + 4y – 8 = 0 into standard form

Application – You Try A skating park has a track shaped like an ellipse. If the length of the track is 58m and the width of the track is 38m, find the equation of the ellipse.

Eccentricity To measure the ovalness of an ellipse, we use eccentricity: