Chapter 9. Analytic Geometry

9.3 The Ellipse 1 Chapter 9. Analytic Geometry

9.3. The Ellipse

Note. In preparation for this section, you may need to review Sections A.5, 1.1, 1.2, and 2.5.

Deﬁnition. An ellipse is the collection of all points in the plane the sum of whose distances from two ﬁxed points, called the foci, is a constant.

Call the foci F1 and F2. The line line containing F1 and F2 is the major axis. The midpoint of the line segment joining the foci is the center of the ellipse. The line through the center and perpendicular to the major axis is the minor axis. The two points of intersection of the ellipse and the major axis are the vertices, V1 and V2, of the ellipse. The distance from one vertex to the other is the length of the major axis.

Note. From the deﬁnition, it follows that we can draw an ellipse by driving two nails into the plane, looping a piece of string around the nails, and then putting a pencil in the loop. By then pulling the loop tight and and drawing, we get an ellipse. 9.3 The Ellipse 2

Figure 17 Page 625

Note. Suppose the foci of an ellipse lie at F1 =(−c, 0) and F2 =(c, 0), that P =(x, y) is a point on the ellipse, and that the sum of the distances from P to F1 and F2 is 2a.

Figure 18 Page 625 9.3 The Ellipse 3

Then by the deﬁnition of an ellipse,

d(F1,P)+d(F2,P)=2a p(x − (−c))2 +(y − 0)2 + p(x − c)2 +(y − 0)2 =2a p(x + c)2 + y2 + p(x − c)2 + y2 =2a p(x + c)2 + y2 =2a − p(x − c)2 + y2 (x + c)2 + y2 =4a2 − 4ap(x − c)2 + y2 +(x − c)2 + y2 x2 +2cx + c2 + y2 =4a2 − 4ap(x − c)2 + y2 +x2 − 2cx + c2 + y2 4cx − 4a2 = −4ap(x − c)2 + y2 cx − a2 = −ap(x − c)2 + y2 (cx − a2)2 = a2[(x − c)2 + y2] c2x2 − 2a2cx + a4 = a2(x2 − 2cx + c2 + y2) (c2 − a2)x2 − a2y2 = a2c2 − a4 (a2 − c2)x2 + a2y2 = a2(a2 − c2) x2 y2 + =1. a2 a2 − c2 We deﬁne b>0 such that b2 = a2 − c2 to get x2 y2 + =1. a2 b2 9.3 The Ellipse 4

Theorem. Equation of an Ellipse; Center at (0, 0); Foci at (±c, 0); Major Axis along the x-Axis. An equation of the ellipse with center at (0, 0) and foci at (−c, 0) and (c, 0) is x2 y2 + =1, where a>b>0 and b2 = a2 − c2. a2 b2 The major axis is the x-axis. The vertices are at (−a, 0) and (a, 0).

Figure 19 Page 626

Example. Page 633 numbers 16 and 18. 9.3 The Ellipse 5

Note. If we interchange the values of x and y (and hence have center (0, 0) and foci on the y-axis), then we get the following.

Theorem. Equation of an Ellipse; Center at (0, 0); Foci at (0, ±c); Major Axis along the y-Axis. An equation of the ellipse with center at (0, 0) and foci at (0, −c) and (0,c)is x2 y2 + =1, where a>b>0 and b2 = a2 − c2. b2 a2 The major axis is the y-axis. The vertices are at (0, −a) and (0,a).

Example. Page 633 numbers 24 and 32.

Note. We can translate the ellipses with centers (0, 0) to create more general equations for an ellipse.

Center Major Axis Foci Vertices Equation (x − h)2 (y − k)2 (h, k) Parallel to x-axis (h + c, k)(h + a, k) + =1, a2 b2 (h − c, k)(h − a, k) a>band b2 = a2 − c2 (x − h)2 (y − k)2 (h, k) Parallel to y-axis (h, k + c)(h, k + a) + =1, b2 a2 (h, k − c)(h, k − a) a>band b2 = a2 − c2

Examples. Page 633 number 44, Page 634 number 62. 9.3 The Ellipse 6

Note. According to Kepler’s First Law of Planetary Motion, the planets orbit the Sun in elliptical orbits with the Sun at one focus. If a room is made in the shape of an ellipse, then sound waves emitted at one focus will reﬂect oﬀ of the walls and converge at the other focus. This is the principle behind various “whispering galleries” (see page 632).

Example. Page 635 number 80 (see the instructions on page 634).