4 minutes Warm-Up Write the standard equation of the circle with the given radius and center. 1) 9; (0,0) 2) 1; (0,5) 3) 4; (-8,-1) 4) 5; (4,2)

9.4.1 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, F1 and F2, called the foci, is a constant. P F1 F2 F1P + F2P = 2a

Standard Equation of an Ellipse Horizontal Major Axis: F1(–c, 0) F2 (c, 0) y x V1(–a, 0) V2 (a, 0) (0, b) (0, –b) O x2 a2 y2 b2 + = 1 a2 > b2 a2 – b2 = c2 length of major axis: 2a length of minor axis: 2b

Standard Equation of an Ellipse Vertical Major Axis: x2 b2 y2 a2 + = 1 F1(0, –c) F2 (0, c) y x V1(0, –a) V2 (0, a) (b, 0) (–b, 0) O a2 > b2 a2 – b2 = c2 length of major axis: 2a length of minor axis: 2b

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. -8 -6 -4 -2 2 4 6 8 length of major axis: 2a 2a = 20, so a = 10 a2 – b2 = c2 102 – b2 = 82 b2 = 100 - 64 b2 = 36, so b = 6 x2 100 y2 36 + = 1

Example 2 Find the vertices and co-vertices of the ellipse. x2 16 y2 49 + = 1 vertices: (0,7) and (0,-7) co-vertices: (4,0) and (-4,0)

Example 3 Write the standard equation of the ellipse. x2 16 y2 64 + length of major axis: 2a 2a = 16, so a = 8 -8 -6 -4 -2 2 4 6 8 length of minor axis: 2b 2b = 8, so b = 4 x2 16 y2 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.

Homework p.591 #11,13,21,27,29,41,43,45 Quiz Tomorrow (9.1-9.3)

5 minutes Warm-Up Write the standard equation for an ellipse with foci at (-5,0) and (5,0) and with a major axis of 18. Sketch the graph.

9.4.2 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

Standard Equation of a Translated Ellipse Horizontal Major Axis: (x – h)2 a2 (y – k)2 b2 + = 1 a2 > b2 a2 – b2 = c2 length of major axis: 2a length of minor axis: 2b

Standard Equation of a Translated Ellipse Vertical Major Axis: (x – h)2 b2 (y – k)2 a2 + = 1 a2 > b2 a2 – b2 = c2 length of major axis: 2a length of minor axis: 2b

Example 1 An ellipse is defined by the equation 4x2 + 9y2 – 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. 4x2 – 16x + 9y2 + 18y = 11 4(x2 – 4x) + 9(y2 + 2y) = 11 4(x2 – 4x + 4) + 9(y2 + 2y + 1) = 11 + 4(4) + 9(1) 4(x – 2)2 + 9(y + 1)2 = 36

Example 1 An ellipse is defined by the equation 4x2 + 9y2 – 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. -6 -4 -2 2 4 6 center: (2,-1) a2 = 9, so a = 3 vertices: (-1,-1) and (5,-1) b2 = 4, so b = 2 co-vertices: (2,1) and (2,-3) a2 – b2 = c2 9 - 4 = c2

Practice Write the standard equation for the ellipse 9x2 + 16y2 – 36x – 64y – 44 = 0. Identify the center, vertices, co-vertices, and foci.

Homework p.591 #19,23,25,31,33,37,51,55