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Ellipses Date: ____________

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Ellipses Standard Equation of an Ellipse Center at (0,0) x2x2 a2a2 y2y2 b2b2 += 1 y x (–a, 0) (a, 0) (0, b) (0, –b) O

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Horizontal Major AxisVertical Major Axis Vertices Co-Vertices Vertices Co-Vertices a 2 > b 2 a 2 < b 2

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An ellipse is the set of all points P in a plane such that the sum of the distances from P to two fixed points, F 1 and F 2, called the foci, is a constant. P F1F1 F2F2 F 1 P + F 2 P = 2a P P 2a For example,

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Horizontal Major Axis: a 2 > b 2 a 2 – b 2 = c 2 x2x2 a2a2 y2y2 b2b2 += 1 F 1 (–c, 0) F 2 (c, 0) y x (–a, 0) (a, 0) (0, b) (0, –b) O length of major axis: 2a length of minor axis: 2b Distance from midpoint and foci: c

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F 2 (0, –c) F 1 (0, c) y x (0, –b) (0, b) (a, 0)(–a, 0) O Vertical Major Axis: b 2 > a 2 x2x2 a2a2 y2y2 b2b2 += 1 length of major axis: 2b length of minor axis: 2a Distance from midpoint and foci: c b 2 – a 2 = c 2

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Write an equation of an ellipse in standard form with the center at the origin and with the given vertex and co-vertex. (4,0), (0,3) Vertices : (4,0) (-4,0) Co-Vertices: (0,3) (0,-3) So a =4 a² = 16 So b = 3 b² = 9 x 2 16 y2y2 9 += 1

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Find an equation of an ellipse for the given height and width with the center at (0,0) h = 32 ft, w = 16 ft 32 ft Distance b is from the center is 16 b = ft Distance a is from the center is 8 a = 8 x 2 64 y2y = 1 a² = 64 b² = 256

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Find the foci and graph the ellipse. x y x 2 25 y2y2 9 += 1 25 – 9 = c 2 16 = c 2 ±4 = c (–4, 0) (4, 0) a 2 = 25 a = ± 5 b 2 = 9 b = ± 3 (–5, 0) (5, 0) (0, 3) (0,-3)

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Graph the ellipse. Find the foci. x y x 2 9 y2y2 25 += 1 (–3, 0)(3, 0) (0, 5) (0,-5) a 2 = 9 a = ± 3 b 2 = 25 b = ± 5 25 – 9 = c 2 16 = c 2 ±4 = c b 2 – a 2 = c 2 (0,4) (0,-4)

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Write an equation of an ellipse for the given foci and co-vertices. Foci: (±5,0), co-vertices: (0,±8) Since c = 5 and b = 8 c² = 25 and b² = 64 Horizontal axis a 2 – b 2 = c 2 a 2 – 64 = a 2 = 89 x2x2 89 y2y = 1 x2x2 a2a2 y2y2 b2b2 +

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Translated Ellipses Standard Equation of an Ellipse Center at (h,k) 9.4 Ellipses (x – h) 2 a2a2 (y – k) 2 b2b2 += 1 y x (h–a, k) (h+a, k) (h, k+b) (h, k–b) (h,k)

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Write an equation of the translation. Length of major axis is 2a 2a = 12 b 2 = 16 (x – 2) 2 36 (y + 5) = 1 Center = (2,-5) Horizontal major axis of length 12, minor axis of length 8. a = 6 Length of minor axis is 2b 2b = 8 b = 4 a 2 = 36 h = 2 k = -5 (x – h) 2 a2a2 (y – k) 2 b2b2 += 1

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Find the foci for the ellipse. 4x 2 + 9y 2 – 16x +18y – 11 = 0 4x 2 – 16x + 9y y = (x 2 – 4x + ____) + 9(y 2 + 2y + ___) = (x – 2) 2 + 9(y + 1) 2 = (x – 2) 2 9 (y + 1) = 1

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a 2 = 9b 2 = 4 Center = (2,-1) (x – 2) 2 9 (y + 1) = 1 a 2 > b 2 Horizontal Axis a 2 – b 2 = c 2 9 – 4 = c 2 5 = c 2 ±2.2 ≈ c Foci = ( ,-1) Foci = (2 – 2.2,-1) Foci = (4.2,-1) and = (-0.2, -1)

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