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Ellipses Date: ____________ Ellipses Standard Equation of an Ellipse Center at (0,0) y x2 + 2 a y2 b2 (–a, 0) (0, b) =1 O (0, –b) (a, 0) x Horizontal Major Axis Co-Vertices Vertical Major Axis Vertices Vertices a2 > b2 Co-Vertices a2 < b2 For example, 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, F1 and F2, called the foci, is a constant. P P P F1 F2 2a F1P + F2P = 2a Horizontal Major Axis: x2 a2 + y2 b2 y =1 (–a, 0) a 2 > b2 a 2 – b2 = c2 (0, b) O F1(–c, 0) (0, –b) length of major axis: 2a length of minor axis: 2b Distance from midpoint and foci: c (a, 0) x F2 (c, 0) Vertical Major Axis: x2 + 2 a y2 b2 =1 b2 > a2 y F1 (0, c) (–a, 0) O b2 – a2 = c 2 length of major axis: 2b length of minor axis: 2a (0, b) F2(0, –c) Distance from midpoint and foci: c (a, 0) x (0, –b) 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) Co-Vertices: (0,3) (0,-3) Vertices : (4,0) (-4,0) So a = 4 So b = 3 a² = 16 x2 16 b² = 9 + y2 9 =1 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 16 ft Distance a is from the center is 8 a=8 a² = 64 b = 16 b² = 256 x2 64 + y2 256 =1 Find the foci and graph the ellipse. x2 25 a2 = 25 a = ±5 + y2 9 =1 y b2 = 9 b = ±3 25 – 9 = c2 16 = c2 ±4 = c (0, 3) (–5, 0) (–4, 0) (5, 0) (4, 0) (0,-3) x Graph the ellipse. Find the foci. x2 9 y2 25 + a2 = 9 a = ±3 =1 y b2 = 25 b = ±5 b2 – a2 = c 2 25 – 9 = c2 16 = c2 ±4 = c (0, 5) (0,4) (–3, 0) (3, 0) x (0,-4) (0,-5) Write an equation of an ellipse for the given foci and co-vertices. Foci: (±5,0), co-vertices: (0,±8) Horizontal axis Since c = 5 and b = 8 – = a2 – 64 = 25 + 64 + 64 a2 = 89 a2 b2 c2 c² = 25 and b² = 64 x2 + 2 a x2 89 + y2 b2 y2 64 =1 =1 9.4 Ellipses Translated Ellipses Standard Equation of an Ellipse Center at (h,k) (x – h)2 (y – k)2 + =1 2 2 a b (h–a, k) y (h, k+b) (h+a, k) (h,k) (h, k–b) x Write an equation of the translation. Center = (2,-5) h=2 k = -5 Horizontal major axis of length 12, minor axis of length 8. Length of major axis is 2a Length of minor axis is 2b 2b = 8 2a = 12 b=4 a=6 a2 = 36 b2 = 16 2 2 2 2 (x – 2) (y + 5) (x – h) (y – k) + =1 + = 1 36 16 a2 b2 Find the foci for the ellipse. 4x2 + 9y2 – 16x +18y – 11 = 0 4x2 – 16x + 9y2 + 18y = 11 4 1 =11+16+9 4(x2 – 4x + ____) + 9(y2 + 2y + ___) 4(x – 2)2 + 9(y + 1)2 = 36 36 (x – 2)2 (y + 1)2 + =1 9 4 (x – 2)2 9 + (y + 1)2 4 =1 Foci = (2 + 2.2,-1) Foci = (2 – 2.2,-1) Center = (2,-1) a2 = 9 a2 > b2 a2 – b 2 = c2 b2 = 4 Horizontal Axis 9 – 4 = c2 5 = c2 ±2.2 ≈ c Foci = (4.2,-1) and = (-0.2, -1)