Ellipses Objectives: Write the standard equation for an ellipse given sufficient information Given an equation of an ellipse, graph it and label the center, vertices, co-vertices, and foci

Definition of Ellipse 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

Standard Equation of an Ellipse 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 V 1 (–a, 0) V 2 (a, 0) (0, b) (0, –b) O length of major axis: 2a length of minor axis: 2b

Standard Equation of an Ellipse Vertical Major Axis: a 2 > b 2 a 2 – b 2 = c 2 x2x2 b2b2 y2y2 a2a2 += 1 length of major axis: 2a length of minor axis: 2b F 1 (0, –c) F 2 (0, c) y x V 1 (0, –a) V 2 (0, a) (b, 0)(–b, 0) O

Example 1 Write the standard equation for an ellipse with foci at (-8,0) and (8,0) and with a major axis of 20. Sketch the graph length of major axis: 2a 2a = 20,so a = 10 a 2 – b 2 = c – b 2 = 8 2 b 2 = b 2 = 36,so b = 6 x2x2 100 y2y2 36 += 1

Example 2 Find the vertices and co-vertices of the ellipse. x2x2 16 y2y2 49 += 1 vertices:(0,7) and (0,-7) co-vertices:(4,0) and (-4,0)

Example 3 Write the standard equation of the ellipse length of major axis: 2a 2a = 16,so a = 8 length of minor axis: 2b 2b = 8,so b = 4 x2x2 16 y2y2 64 += 1

Practice Write the standard equation for an ellipse with foci at (5,0) and (-5,0) and with vertices at (9,0) and (-9,0). Sketch the graph.

Standard Equation of a Translated Ellipse a 2 > b 2 a 2 – b 2 = c 2 (x – h) 2 a2a2 (y – k) 2 b2b2 += 1 Horizontal Major Axis: length of major axis: 2a length of minor axis: 2b

Standard Equation of a Translated Ellipse a 2 > b 2 a 2 – b 2 = c 2 Vertical Major Axis: (x – h) 2 b2b2 (y – k) 2 a2a2 += 1 length of major axis: 2a length of minor axis: 2b

Example 1 An ellipse is defined by the equation 4x 2 + 9y 2 – 16x + 18y = 11. Write the standard equation and identify the coordinates of the center, vertices, co-vertices, and foci. Sketch the graph of the ellipse. 4(x 2 – 4x) + 9(y 2 + 2y) = 11 4x 2 – 16x + 9y y = 11 4(x 2 – 4x + 4) + 9(y 2 + 2y + 1) = (4) + 9(1) 4(x – 2) 2 + 9(y + 1) 2 = 36

Example 1 An ellipse is defined by the equation 4x 2 + 9y 2 – 16x + 18y = 11. Write the standard equation and identify the coordinates of the center, vertices, co-vertices, and foci. Sketch the graph of the ellipse. center: (2,-1) a 2 = 9, so a = vertices: (-1,-1) and (5,-1) b 2 = 4, so b = 2 co-vertices: (2,1) and (2,-3) a 2 – b 2 = c = c 2

Practice Write the standard equation for the ellipse 9x y 2 – 36x – 64y – 44 = 0. Identify the center, vertices, co-vertices, and foci. Center (2,2) Vertices (6, 2) (-2, 2) Co vertices (2, 5) (2, -1) Foci

Homework Pg 646 #1-29 odd, 44 Quiz on circles and ellipses next class