The Ellipse

Definition of an Ellipse An ellipse is the set of all points in a plane the sum of whose distances from two fixed points, is constant. These two fixed points are called the foci. The midpoint of the segment connecting the foci is the center of the ellipse. F1 F2 P

Standard Forms of the Equations of an Ellipse The standard form of the equation of an ellipse with center at the origin, and major and minor axes of lengths 2a and 2b (where a and b are positive, and a2 > b2) is or The vertices are on the major axis, a units form the center. The foci are on the major axis, c units form the center. For both equations, b2 = a2 – c2. (0, b) (0, -b) (0, a) (0, -a) (0, c) (0, -c) (0, 0) (a, 0) (-a, 0) (c, 0) (-c, 0) (b, 0) (-b, 0)

Text Example Graph and locate the foci: 25x2 + 16y2 = 400. Solution We begin by expressing the equation in standard form. Because we want 1 on the right side, we divide both sides by 400. b2 = 16. This is the smaller of the two numbers in the denominator. a2 = 25. This is the larger of the two numbers in the denominator. The equation is the standard form of an ellipse’s equation with a2 = 25 and b2 = 16. Because the denominator of the y2 term is greater than the denominator of the x2 term, the major axis is vertical.

Text Example cont. Solution Based on the standard form of the equation, we know the vertices are (0, -a) and (0, a). Because a2 =25, a = 5. Thus, the vertices are (0, -5) and (0, 5). Solution Now let us find the endpoints of the horizontal minor axis. According to the standard form of the equation, these endpoints are (-b, 0) and (b, 0). Because b2 = 16, b = 4. Thus, the endpoints are (-4, 0) and (4, 0). (0, 5) (0, -5) (0, 3) (0, -3) (0, 0) (4, 0) (-4, 0) Finally, we find the foci, which are located at (0, -c) and (0, c). Because b2 = a2 – c2, a2 =25, and b2 = 16, we can find c as follows: c2 = a2 – b2 = 25 – 16 = 9. Because c2 = 9, c = 3. The foci, (0, -c) and (0, c), are located at (0, -3) and (0, 3). Sketch the graph shown by locating the endpoints on the major and minor axes.

Standard Forms of Equations of Ellipses Centered at (h,k) (h, k – a) (h, k + a) (h, k – c) (h, k + c) Parallel to the y-axis, vertical (h, k) b2 > a2 and a2 = b2 – c2 (h – a, k) (h + a, k) (h – c, k) (h + c, k) Parallel to the x-axis, horizontal a2 > b2 and b2 = a2 – c2 Vertices Foci Major Axis Center Equation (h, k) Major axis x y Focus (h – c, k) Focus (h + c, k) Vertex (h – a, k) Vertex (h + a, k)

Text Example Graph: Where are the foci located? Solution In order to graph the ellipse, we need to know its center (h, k). In the standards forms of centered at (h, k), h is the number subtracted from x and k is the number subtracted from y. This is (x – h)2 with h = 1. This is (y – k)2 with h = -2. We see that h = 1 and k = -2. Thus, the center of the ellipse, (h, k) is (1, -2). We can graph the ellipse by locating endpoints on the major and minor axes. To do this, we must identify a2 and b2. a2 = 9. This is the larger of the two numbers in the denominator. b2 = 4. This is the smaller of the two numbers in the denominator.

Text Example cont. Solution The larger number is under the expression involving y. This means that the major axis is vertical and parallel to the y-axis. Because a2 = 9, a = 3 and the vertices lie three units above and below the center. Also, because b2 = 4, b = 2 and the endpoints of the minor axis lie two units to the right and left of the center. We categorize these observations as follows: -5 -4 -3 -2 -1 1 2 3 4 5 (-1, -2) (3, -2) (1, -5) (1, 1) (1, -2) Endpoints of Minor Axis Vertices Center (1 - 2, -2) = (3, -2) (1, -2 - 3) = (1, 1) (1 + 2, -2) = (3, -2) (1, -2 + 3) = (1, 1) (1, -2) With b2 = a2 – c2, we have 4 = 9 – c2, and c2 = 5. So the foci are located 5 units above and below the center, at (1,-2+ 5) and (1, -2– 5 ).

Example Find the standard form of the equation of the ellipse centered at the origin with Foci (0,-3),(0,3) and vertices (0,-5), (0,5) Solution: a = 5 and c = 3

The Ellipse