CHAPTER 9 CONIC SECTIONS 9 2 Ellipses Definition

CHAPTER 9: CONIC SECTIONS 9. 2. Ellipses Definition of Ellipse An ellipse is a locus of all points (x, y) 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 F 1 F 2 F 1 P + F 2 P = 2 a Copyright © by Holt, Rinehart and Winston. All Rights Reserved.

Parts of an ellipse Center (h, k) Major Axes: Co-vertex focus Vertex Minor Axes: ) h, k ( r nte e C Major axis Minor axis Co-vertex Vertices: Co-vertices: Foci: There are TWO cases of an ellipse: Horizontal major axes and Vertical major axes Copyright © by Holt, Rinehart and Winston. All Rights Reserved.

Properties and Equation of an ellipse Horizontal Major Axis and C(0, 0): y a 2 > b 2 V 1(–a, 0) (0, b) a 2 – b 2 = c 2 x 2 + 2 a y 2 b 2 O =1 F 1(–c, 0) (0, –b) V 2 (a, 0) x F 2 (c, 0) major axis = 2 a minor axis = 2 b Copyright © by Holt, Rinehart and Winston. All Rights Reserved.

Rules and Properties Horizontal Major Axis and C(h, k): (x – h)2 (y – k)2 + =1 2 2 a b a 2 > b 2 a 2 – b 2 = c 2 Copyright © by Holt, Rinehart and Winston. All Rights Reserved.

Properties and Equation of an ellipse Vertical Major Axis and C(0, 0): a 2 > y b 2 F 2 (0, c) a 2 – b 2 = c 2 x 2 + 2 b y 2 a 2 V 2 (0, a) =1 major axis = 2 a minor axis = 2 b (–b, 0) O F 1(0, –c) (b, 0) x V 1(0, –a) Copyright © by Holt, Rinehart and Winston. All Rights Reserved.

Rules and Properties Vertical Major Axis and C(h, k): (x – h)2 (y – k)2 + =1 2 2 b a a 2 > b 2 a 2 – b 2 = c 2 Copyright © by Holt, Rinehart and Winston. All Rights Reserved.

ECCENTRICITY OF AN ELLIPSE The early Greek astronomers thought that the planets moved in circular orbits about an unmoving earth. In the 17 th century, Johannes Kepler discovered that each planet travels around the sun in an elliptical orbit Copyright © by Holt, Rinehart and Winston. All Rights Reserved.

ECCENTRICITY OF AN ELLIPSE One of the reasons it was difficult to detect that orbits are elliptical is that the foci of the planetary orbits are relatively close to the center, making the ellipse nearly circular. To measure the ovalness of an ellipse, we use the concept of eccentricity. DEFINITION: The eccentricity e of an ellipse is given by the ratio e = c/a e e 1 Copyright © by Holt, Rinehart and Winston. All Rights Reserved.

PRACTICE EX. 1: Write equations of ellipses graphed in the coordinate plane Copyright © by Holt, Rinehart and Winston. All Rights Reserved.

PRACTICE EX. 2: Sketch the graph of each ellipse. Identify the center, the vertices, the co-vertices, and the foci for each ellipse. Copyright © by Holt, Rinehart and Winston. All Rights Reserved.

PRACTICE EX. 3: Find the coordinates of the center and vertices of an ellipse. Graph the ellipse. (x – 2)2 (y – 1)2 + =1 16 9 center: (2, 1) vertices: (– 2, 1), (6, 1) Copyright © by Holt, Rinehart and Winston. All Rights Reserved.

PRACTICE EX. 4: Find the coordinates of the co-vertices, and foci of an ellipse. Graph the ellipse. (x – 2)2 (y – 1)2 + =1 16 9 co-vertices: (2, 4), (2, – 2) foci: (2 – 7 , 1), (2 + 7 , 1) Copyright © by Holt, Rinehart and Winston. All Rights Reserved.

PRACTICE EX. 5: Graph the ellipse. 9 x 2 + 16 y 2 – 36 x – 32 y – 92 = 0 standard form: (x – 2)2 (y – 1)2 + =1 16 9 Copyright © by Holt, Rinehart and Winston. All Rights Reserved.