10.3 Ellipses.

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10.3 Ellipses

10.3 Ellipses Center point (h,k) Focus point a b F F c V V a Minor axis Major axis An ellipse is the set of all points (x,y), the sum of whose distances from two distinct points (foci) is constant. a2 = b2 + c2

Standard Equation of an Ellipse Horz. Major axis Vert. Major axis (h,k) is the center point. The foci lie on the major axis, c units from the center. c is found by c2 = a2 - b2 Major axis has length 2a and minor axis has length 2b.

Sketch and find the Vertices, Foci, and Center point. x2 + 4y2 + 6x - 8y + 9 = 0 First, write the equation in standard form. (x2 + 6x + ) + 4(y2 - 2y + ) = -9 (x2 + 6x + 9) + 4(y2 - 2y + 1) = -9 + 9 + 4 (x + 3)2 + 4(y - 1)2 = 4 C (-3,1) V (-1,1) (-5,1)

c2 = a2 - b2 C (-3,1) c2 = 4 - 1 V (-1,1) (-5,1) Foci are:

Eccentricity e of an ellipse measures the ovalness of the ellipse. e = c/a In the last example, what is the eccentricity? The smaller or closer to 0 that the eccentricity is, the more the ellipse looks like a circle. The closer to 1 the eccentricity is, the more elongated it is.

Find the center, vertices, and foci of the ellipse given by 4x2 + y2 - 8x + 4y - 8 = 0 First, put this equation in standard form. 4(x2 - 2x + 1) + ( y2 + 4y + 4) = 8 + 4 + 4 4(x - 1)2 + (y + 2)2 = 16 C( , ) a = b = c = Vertices ( , ) ( , ) Foci ( , ) ( , ) e = Sketch it.

Assignment: 1-6 all, 7 - 29 odd