Ellipses Date: ____________

Ellipses Standard Equation of an Ellipse Center at (0,0) x2 a2 y2 b2 + = 1 (a, 0) x O (0, –b)

Horizontal Major Axis Vertical Major Axis Co-Vertices Vertices a2 < b2 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: y x2 a2 y2 b2 + = 1 (0, b) (a, 0) (–a, 0) a2 > b2 a2 – b2 = c2 x O F1(–c, 0) F2 (c, 0) (0, –b) length of major axis: 2a length of minor axis: 2b Distance from midpoint and foci: c

length of major axis: 2b length of minor axis: 2a Vertical Major Axis: y x2 a2 y2 b2 + = 1 (0, b) F1 (0, c) b2 > a2 (–a, 0) (a, 0) x O b2 – a2 = c2 F2(0, –c) length of major axis: 2b length of minor axis: 2a (0, –b) Distance from midpoint and foci: c

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) Co-Vertices: (0,3) (-4,0) (0,-3) So a = 4 So b = 3 a² = 16 b² = 9 x2 16 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 x2 64 y2 256 + = 1 a = 8 a² = 64 b = 16 b² = 256

Find the foci and graph the ellipse. x2 25 y2 9 + = 1 x y a2 = 25 b2 = 9 a = ±5 b = ±3 (0, 3) 25 – 9 = c2 (–5, 0) (–4, 0) (5, 0) 16 = c2 (4, 0) ±4 = c (0,-3)

Graph the ellipse. Find the foci. x2 9 y2 25 + = 1 x y a2 = 9 b2 = 25 (0, 5) a = ±3 b = ±5 (0,4) (–3, 0) (3, 0) b2 – a2 = c2 25 – 9 = c2 (0,-4) 16 = c2 ±4 = c (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 c² = 25 and b² = 64 x2 a2 y2 b2 + = 1 a2 – b2 = c2 a2 – 64 = 25 + 64 + 64 x2 89 y2 64 + = 1 a2 = 89

Translated Ellipses Standard Equation of an Ellipse Center at (h,k) (x – h)2 a2 (y – k)2 b2 + = 1 y (h, k+b) (h+a, k) (h,k) (h, k–b) (h–a, k) 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 2a = 12 2b = 8 a = 6 b = 4 a2 = 36 b2 = 16 (x – 2)2 36 (y + 5)2 16 + = 1 (x – h)2 a2 (y – k)2 b2 + = 1

Find the foci for the ellipse. 4x2 + 9y2 – 16x +18y – 11 = 0 4x2 – 16x + 9y2 + 18y = 11 4(x2 – 4x + ____) + 9(y2 + 2y + ___) =11 4 1 +16 +9 4(x – 2)2 + 9(y + 1)2 = 36 36 (x – 2)2 9 (y + 1)2 4 + = 1

(x – 2)2 9 (y + 1)2 4 = 1 Foci = (2 + 2.2,-1) Foci = (2 – 2.2,-1) Center = (2,-1) Foci = (4.2,-1) and = (-0.2, -1) a2 = 9 b2 = 4 a2 > b2 Horizontal Axis a2 – b2 = c2 9 – 4 = c2 5 = c2 ±2.2 ≈ c