Copyright © 2011 Pearson Education, Inc. The Ellipse and the Circle Section 7.2 The Conic Sections

7.2 Copyright © 2011 Pearson Education, Inc. Slide 7-3 An ellipse is the set of points in a plane such that the sum of their distances from two fixed points is a constant. Each fixed point is called a focus (plural: foci) of the ellipse. Definition: Ellipse

7.2 Copyright © 2011 Pearson Education, Inc. Slide 7-4 An ellipse centered at the origin has foci at (c, 0) and (– c, 0), and y-intercepts (0, b) and (0, – b), where c > 0 and b > 0. The line segment is the major axis, and the line segment is the minor axis. For any ellipse, the major axis is longer than the minor axis and the foci are on the major axis. The center of an ellipse is the midpoint of the major (or minor) axis. The vertices of an ellipse are the endpoints of the major axis. The Equation of the Ellipse

7.2 Copyright © 2011 Pearson Education, Inc. Slide 7-5 Let a be the distance between (c, 0) and the y-intercept (0, b). The sum of the distances from the two foci to (0, b) is 2a. So for any point (x, y) on the ellipse, the distance from (x, y) to (c, 0) plus the distance from (x, y) to (– c, 0) is equal to 2a. The Equation of the Ellipse

7.2 Copyright © 2011 Pearson Education, Inc. Slide 7-6 The Equation of the Ellipse

7.2 Copyright © 2011 Pearson Education, Inc. Slide 7-7 The equation of an ellipse centered at the origin with foci (c, 0) and (– c, 0) and y-intercepts (0, b) and (0, – b) is The sum of the distances from any point on the ellipse to the two foci is 2a. Theorem: Equation of an Ellipse with Center (0, 0) and Horizontal Major Axis

7.2 Copyright © 2011 Pearson Education, Inc. Slide 7-8 The equation of an ellipse centered at the origin with foci (0, c) and (0, – c) and x-intercepts (b, 0) and (– b, 0) is The sum of the distances from any point on the ellipse to the two foci is 2a. Theorem: Equation of an Ellipse with Center (0,0) and Vertical Major Axis

7.2 Copyright © 2011 Pearson Education, Inc. Slide 7-9 Consider the ellipse with foci (c, 0) and (– c, 0) and equation where a > b > 0. If y = 0 in this equation, then x = ±a. So the vertices (or x-intercepts) of the ellipse are (a, 0) and (– a, 0), and a is the distance from the center to a vertex. The distance from a focus to an endpoint of the minor axis is a also. So in any ellipse the distance from the focus to an endpoint of the minor axis is the same as the distance from the center to a vertex. The Equation of the Ellipse

7.2 Copyright © 2011 Pearson Education, Inc. Slide 7-10 For a > b > 0, the graph of is an ellipse centered at the origin with a horizontal major axis, x-intercepts (a, 0) and (– a, 0), and y-intercepts (0, b) and (0, – b). The foci are on the major axis and are determined by a 2 = b 2 + c 2 or c 2 = a 2 – b 2. Remember that when the denominator for x 2 is larger than the denominator for y 2, the major axis is horizontal. For a > b > 0, the graph of is an ellipse centered at the origin with a vertical major axis, x-intercepts (b, 0) and (– b, 0), and y-intercepts (0, a) and (0, – a). The foci are on the major axis and are determined by a 2 = b 2 + c 2 or c 2 = a 2 – b 2. When the denominator for y 2 is larger than the denominator for x 2, the major axis is vertical. Graphing an Ellipse Centered at the Origin

7.2 Copyright © 2011 Pearson Education, Inc. Slide 7-11 Although an ellipse is not the graph of a function, its graph can be translated in the same manner. The graphs of both equations have the same size and shape, but the graph of the first equation is centered at (h, k) rather than the origin. So the graph of the first equation is obtained by translating the graph of the second equation horizontally h units and vertically k units. Translations of Ellipses

7.2 Copyright © 2011 Pearson Education, Inc. Slide 7-12 A point (x, y) is on a circle with center (h, k) and radius r if and only if If we square both sides of this equation, we get the standard equation of a circle. The Circle

7.2 Copyright © 2011 Pearson Education, Inc. Slide 7-13 A circle is a set of all points in a plane such that their distance from a fixed point (the center) is a constant (the radius). Definition: Circle

7.2 Copyright © 2011 Pearson Education, Inc. Slide 7-14 The standard equation of a circle with center (h, k) and radius r (r > 0) is (x – h) 2 + (y – k) 2 = r 2. Theorem: Standard Equation of a Circle

7.2 Copyright © 2011 Pearson Education, Inc. Slide 7-15 The equation (x – h) 2 + (y – k) 2 = s is a circle of radius if s > 0. If s = 0, only (h, k) satisfies (x – h) 2 + (y – k) 2 = 0 and the point (h, k) is a degenerate circle. If s < 0, then no ordered pair satisfies (x – h) 2 + (y – k) 2 = s. If h and k are zero, then we get the standard equation of a circle centered at the origin, x 2 + y 2 = r 2. The Circle

7.2 Copyright © 2011 Pearson Education, Inc. Slide 7-16 A circle is an ellipse in which the two foci coincide at the center. If the foci are identical, then a = b and the equation for an ellipse becomes the equation for a circle with radius a. The eccentricity e of an ellipse is defined by e = c/a, where c is the distance from the center to a focus and a is one-half the length of the major axis. Since 0 < c < a, we have 0 < e < 1. For an ellipse that appears circular, the foci are close to the center and c is small compared with a. So its eccentricity is near 0. For an ellipse that is very elongated, the foci are close to the vertices and c is nearly equal to a. So its eccentricity is near 1. Applications