Notes 10-3: Ellipses I. Ellipse- Definition and Vocab An ellipse is the set of points P(x, y) in a plane such that the sum of the distances from any point P on the ellipse to two fixed points F 1 and F 2, called the foci (singular: focus), is the constant sum d = PF 1 + PF 2. This distance d can be represented by the length of a piece of string connecting two pushpins located at the foci. The center is the midpoint of the foci. To help visualize this: https://www.youtube.com/watch?v=7ud8hos-vai Instead of a single radius, an ellipse has two axes. The longer the axis of an ellipse is the major axis and passes through both foci. The endpoints of the major axis are the vertices of the ellipse. The shorter axis of an ellipse is the minor axis. The endpoints of the minor axis are the covertices of the ellipse. The major axis and minor axis are perpendicular and intersect at the center of the ellipse. The center is also the midpoint of the two foci.
So, b 2 + c 2 = a 2, where b is the length of the semi-minor axis, a is the length of the semi-major axis, and c is the distance from the center to one of the foci.
Ex 1: SPACE The graph models the elliptical path of a space probe around two moons of a planet. The foci of the path are the centers of the moons. Find the coordinates of the foci. The center of the ellipse is the origin. a = 1 2 (314) or 157 b = 1 2 (110) or 55 The distance from the origin to the foci is c km. c 2 + b 2 = a 2 c 2 + 55 2 = 157 2 c 147 The foci are at (147, 0) and (-147, 0). II. Standard Form of an Ellipse
Remember that for b 2 + c 2 = a 2, b is the length of the semi-minor axis, a is the length of the semi-major axis, and c is the distance from the center to one of the foci. Example 2
Consider the ellipse graphed at the right. a. Write the equation of the ellipse in standard form. The center of the graph is at (-2, 1). Therefore, h = -2 and k = 1. Since the ellipse s vertical axis is longer than its horizontal axis, a is the distance between points at (-2, 1) and (-2, -3) or 4. The value of b is the distance between points at (-2, 1) and (1, 1) or 3. Therefore, the standard form of the equation is (y - 1) 2 (x - (-2))2 (y - 1)2 2 4 + 2 3 = 1 or 16 b. Find the coordinates of the foci. + (x + 2)2 9 Using the equation c = a 2 - b 2, we find that c = 7. The foci are located on the vertical axis, = 1. 7 units from the center of the ellipse. Therefore, the foci have coordinates (-2, 1-7) and (-2, 1 + 7). Example 3
(y - 2)2 (x + 1)2 For the equation 100 + 64 = 1, find the coordinates of the center, foci, and vertices of the ellipse. Then graph the equation. Determine the values of a, b, c, h, and k. Since a 2 > b 2, a 2 = 100 and b 2 = 64. Therefore, a = 10 and b = 8. c = a 2 - b 2 c = 100-64 or 6 x h = x + 1 y k = y - 2 h = -1 k = 2 Since a 2 is the denominator of the y term, the major axis is parallel to the y-axis. center: (-1, 2) (h, k) foci: (-1, 8) and (-1, -4) (h, k c) major axis vertices: (-1, 12) and (-1, -8) (h, k a) minor axis vertices: (-9, 2) and (7, 2) (h b, k) Graph these ordered pairs. Other points on the ellipse can be found by substituting values for x and y. Complete the ellipse.
III. General Form of an Ellipse Ax 2 + Cx 2 + Dx + Ey + F = 0 ; A and C have the same sign. Example 4 Find the coordinates of the center, the foci, and the vertices of the ellipse with the equation 9x 2 + 16y 2 + 54x 32y 47 = 0. Then graph the equation. First, write the equation in standard form. 9x 2 + 16y 2 + 54x 32y 47 = 0 9(x 2 + 6x +?) + 16(y 2 2y +?) = 47 +? +? 9(x 2 + 6x + 9) + 16(y 2 2y + 1) = 47 + 9(9) + 16(1) CTS 9(x + 3) 2 + 16(y 1) 2 = 144 Factor. (x + 3) 2 16 + (y - 1)2 9 = 1 Divide each side by 144. Since a 2 > b 2, a 2 = 16 and b 2 = 9. Thus, a = 4 and b = 3. Since c 2 = a 2 b 2, c = 7. Since a 2 is the denominator of the x term, the major axis is parallel to the x-axis. center: (-3, 1) (h, k) foci: (-3 7, 1) (h c, k) major axis vertices: (-7, 1), (1, 1) (h a, k) minor axis vertices: (-3, 4), (-3, -2) (h, k b)
Sketch the ellipse. IV. Eccentricity The eccentricity of an ellipse, denoted by e, is a measure that describes the shape of an ellipse. It is defined as. e is always between 0 and 1. Sometimes you may need to know the value of b when you know the values of a and e. In any ellipse, c 2 = a 2 b 2, so b 2 = c 2 a 2. This means that b 2 = a 2 (1 e 2 ).
Example 5 ASTRONOMY The eccentricity of Uranus is 0.047. Its orbit is about 18.3 AU (astronomical units) from the sun at its closest point to the sun. The length of the semi-major axis of the orbit is about 19.21 AU. Sketch a diagram. Find the length of the semi-minor axis of the orbit. We are looking for b. b 2 = a 2 (1 e 2 ) b 2 = (19.21) 2 (1 (.047) 2 ) b 2 =368.20 b = 19.19 AU Find the distance of Uranus from the sun at its farthest point. Aphelion = length of major axis sun to perihelion d = 2(19.19) 18.3 d = 20 Uranus is about 20 AU from the sun at its aphelion.