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10.3 Ellipses

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. a 2 = b 2 + c 2. Standard Equation of an Ellipse.

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10.3 Ellipses

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  1. 10.3 Ellipses

  2. 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

  3. 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.

  4. Sketch and find the Vertices, Foci, and Center point. 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)

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

  6. 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.

  7. 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.

  8. Assignment: 1-6 all, 7 - 29 odd

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