7 2 Notes Ellipses and Circles Objectives Analyze

7. 2 Notes: Ellipses and Circles Objectives: - Analyze the graphs of ellipses and circles - Use equations to identify ellipses and circles

An ellipse is the locus of points in a plane such that the sum of the distances from two fixed points, called foci, is constant. Foci: two fixed points, equidistant from center of an ellipse Major Axis: the segment that contains the foci of the ellipse and has its endpoints on the ellipse. (length = 2 a) Minor Axis: the segment through the center of an ellipse that is perpendicular to the major axis and has endpoints on the ellipse. (length = 2 b)

Center: the midpoint of major and minor axes of an ellipse. Vertices: the endpoints of the major axis of an ellipse. The distance from each vertex to the center is a units. Co-vertices: the endpoints of the minor axis of an ellipse. The distance from each co-vertex to the center is b units. *The distance from the center to each focus is c units.

Example 1: Graph the ellipse given by each equation. a)

Example 1: Graph the ellipse given by each equation. b) 4 x 2 + 24 x + y 2 – 10 y – 3 = 0

The value c represents the distance between one foci and the center of the ellipse. As the foci are moved closer together, c approaches 0. When this happens, the ellipse is a circle and both a and b are equal to the radius of the circle.

Example 3: Graph the circle given by the equation. A) (x – 4)2 + (y + 1) 2 = 4

Example 3: Graph the circle given by the equation. B) x 2 – 4 x + y 2 + 6 y = – 9

Warm-Up Sketch a graph of the ellipse with the equation:

Homework Answers Ellipses WB p. 17 #1 -2, #15 - 18 1. 2. Equation: Foci: Vertices: (-2, 2) (4, 2) Co-Vertices: (1, 0) (1, 4) Equation: Foci: (1, 1) (1, 9) Vertices: (1, 0) (1, 10) Co-Vertcies (-2, 5) (4, 5)

15) 17) center: (0, 0) r = 4 center: (-2, 0) r = 4 16) center (2, 1) r = 3 18) center (0, 1) r = 3

Example 2: A) Write an equation for an ellipse with a major axis from (5, – 2) to (– 1, – 2) and a minor axis from (2, 0) to (2, – 4).

Example 2: B) Write an equation for an ellipse with vertices at (3, – 4) and (3, 6) and foci at (3, 4) and (3, – 2).

Example 2: C) Write an equation for an ellipse with co-vertices at (– 8, 6) and (4, 6) and major axis of length 18.

Example 2: D) Write an equation for the ellipse.

Example 2: E) Write an equation for the ellipse.

Determining Types of Conics Use the characteristics of the standard form equations to help identify the conic. Example 5: Write each equation in standard form. Identify the related conic. a) 9 x 2 + 4 y 2 + 8 y – 32 = 0