SECTION 10.3 The Ellipse 693 The definition contains within it a physical means for drawing an ellipse. Find a piece of string (the length of the string is the constant referred to in the definition). Then take two thumbtacks (the foci) and stick them into a piece of cardboard so that the distance between them is less than the length of the string. Now attach the ends of the string to the thumbtacks and, using the point of a pencil, pull the string taut. See Figure 19. Keeping the string taut, rotate the pencil around the two thumbtacks. The pencil traces out an ellipse, as shown in Figure 19. In Figure 19, the foci are labeled F1 and F .2 The line containing the foci is called 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 V ,2 of the ellipse. The distance from one vertex to the other is the length of the major axis (which is the constant length of the string). The ellipse is symmetric with respect to its major axis, with respect to its minor axis, and with respect to its center. 1 Analyze Ellipses with Center at the Origin With these ideas in mind, we are ready to find the equation of an ellipse in a rectangular coordinate system. First, place the center of the ellipse at the origin. Second, position the ellipse so that its major axis coincides with a coordinate axis, say the x-axis, as shown in Figure 20. If c is the distance from the center to a focus, one focus will be at F c, 0 1 ( ) = − and the other at F c, 0 . 2 ( ) = As we shall see, it is convenient to let a2 denote the constant distance referred to in the definition. Then, if P x y , ( ) = is any point on the ellipse, Figure 19 Ellipse V1 P V2 Center F1 F2 Minor axis Major axis Figure 20 x y P 5 (x, y) d (F2, P) d (F1, P) F1 5 (2c, 0) F2 5 (c, 0) The sum of the distances from P to the foci equals a constant, a2 . Use the Distance Formula. Isolate one radical. Square both sides. Multiply out. Simplify; isolate the radical. Divide both sides by 4. Square both sides again. Multiply out. Rearrange the terms. Multiply both sides by −1; factor out a2 on the right side. [ ] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) + − + + − = − − + − = − − + − = − + − + = − + + − − = − − + = − x cx c y cx a a x c y cx a a x c y cx a a x c y c x acxa ax cxc y c a x a y a c a a c x a y a a c 2 4 4 4 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 2 2 2 2 2 2 2 2 2 2 2 4 2 2 2 2 2 2 2 2 (1) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) + = + + + − + = + + = − − + + + = − − + + − + + + + = − − + d F P d F P a x c y x c y a x c y a x c y x c y a a x c y x c y x cx c y a a x c y , , 2 2 2 4 4 2 4 4 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 To obtain points on the ellipse that are not on the major axis, we must have a c. > To see why, look again at Figure 20. Then d F P d F P d F F a c a c , , , 2 2 1 2 1 2 ( ) ( ) ( ) + > > > The sum of the lengths of any two sides of a triangle is greater than the length of the third side. ( ) ( ) ( ) + = = dFP dFP adFF c , , 2 ; , 2 1 2 1 2
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