The Circle and the Ellipse Section 10.2 The Circle and the Ellipse Copyright ©2013, 2009, 2006, 2005 Pearson Education, Inc.

Objectives Given an equation of a circle, complete the square, if necessary, and then find the center and the radius and graph the circle. Given the equation of an ellipse, complete the square, if necessary, and then find the center, the vertices, and the foci and graph the ellipse.

Circle A circle is the set of all points in a plane that are at a fixed distance from a fixed point (the center) in the plane. Standard Equation of a Circle The standard equation of a circle with center (h, k) and radius r is (x – h)2 + (y – k)2 = r2.

Example For the circle x2 + y2 – 16x + 14y + 32 = 0, find the center and the radius. Then graph the circle. First, we complete the square twice:

Example (continued) Center = (8, –7) Radius = 9

Ellipse An ellipse is the set of all points in a plane, the sum of whose distances from two fixed points (the foci) is constant. The center of an ellipse is the midpoint of the segment between the foci. The major axis of an ellipse is longer than the minor axis.

Standard Equation of an Ellipse with Center at the Origin (Major Axis Horizontal) Vertices: (–a, 0), (a, 0) y-intercepts: (0, –b), (0, b) Foci: (–c, 0), (c, 0) where c2 = a2 – b2

Standard Equation of an Ellipse with Center at the Origin (Major Axis Vertical) Vertices: (0, –a), (0, a) x-intercepts: (–b, 0), (b, 0) Foci: (0, –c), (0, c) where c2 = a2 – b2

Example Find the standard equation of the ellipse with vertices (–5, 0) and (5, 0) and foci (–3, 0) and (3, 0). Then graph the ellipse. Since the foci are on the x-axis and the origin is the midpoint of the segment between them, the major axis is horizontal and (0, 0) is the center of the ellipse. Thus the equation is of the form

Example (continued) Since the vertices are (–5, 0) and (5, 0) and the foci are (–3, 0) and (3, 0), we know that a = 5 and c = 3. These values can be used to find b2. Thus the equation of the ellipse is

Example (continued) Vertices: (–5, 0) and (5, 0) b = 4, so y-intercepts: (0, –4) and (0, 4). We plot the vertices and the y-intercepts and connect the four points with a smooth curve.

Example For the ellipse 9x2 +4y2 = 36, find the vertices and the foci. Then draw the graph. First find the standard form. Thus, a = 3 and b = 2. The major axis is vertical. Vertices: (0, –3) and (0, 3)

Example (continued) Foci: Since b = 2, x-intercepts: (–2, 0) and (2, 0). We plot the vertices and the x-intercepts and connect the four points with a smooth curve.

Standard Equation of an Ellipse with Center at (h, k) (Major Axis Horizontal) Vertices: (h – a, k), (h + a, k) Length of minor axis: 2b Foci: (h – c, k), (h + c, k) where c2 = a2 – b2

Standard Equation of an Ellipse with Center at (h, k) (Major Axis Vertical) Vertices: (h, k – a), (h, k + a) Length of minor axis: 2b Foci: (h, k – c), (h, k + c) where c2 = a2 – b2

Example For the ellipse find the center, the vertices, and the foci. Then draw the graph. First, complete the square twice.

Example (continued) Center is (–3, 1) a = 4 and b = 2 The major axis is vertical so the vertices are 4 units above and below the center: (–3, 1 + 4) (–3, 1 – 4) or (–3, 5) and (–3, –3). We know that c2 = a2 – b2, so c2 = 42 – 22 = 12, so c = . Then the foci are units above and below the center: Since b = 2, endpoints of minor axis are 2 units to the right and left of the center: (–3 + 2, 1) (–3 – 2, 1) or (–1, 1) and (–5, 1).

Example (continued) To graph, we plot the vertices and the endpoints of the minor axis and connect all four points with a smooth curve.

Applications An exciting medical application of an ellipse is a device called a lithotripter. One type of this device uses electromagnetic technology to generate a shock wave to pulverize kidney stones. The wave originates at one focus of an ellipse and is reflected to the kidney stone, which is positioned at the other focus. Recovery time following the use of this technique is much shorter than with conventional surgery and the mortality rate is far lower.

Applications (continued) A room with an ellipsoidal ceiling is known as a whispering gallery. In such a room, a word whispered at one focus can be clearly heard at the other. Whispering galleries are found in the rotunda of the Capitol Building in Washington, D.C., and in the Mormon Tabernacle in Salt Lake City.

Applications (continued) Ellipses have many other applications. Planets travel around the sun in elliptical orbits with the sun at one focus, for example, and satellites travel around the earth in elliptical orbits as well.