Calculus: Analytic Geometry and Calculus, with Vectors

6.3 Ellipses 371

With the aid of Figure 6.31 and these formulas, we find that

(6.334) IY1F,I + IF1F2I + IF2Y2I

= 2IYrFLI + 177F-,.l =

d

cos a= 1Z421 -

From this and (6.33) we obtain the string property (6.32). One reason

for interest in the string property of ellipses lies in the fact that it provides

a mechanicalmethod for drawing ellipses. Let a string of length 2a have

its ends pinned to two points Fl and F2 on a sheet of paper. Using a pencil

point to stretch the string into two straight segments, we can move the

pencil so that its point draws an ellipse having foci at Fl and F2 as the

string slides over the pencil point.

Figure 6.34 shows an ellipse which was drawn with the aid of the string

property, and it also shows some numerical dimensions which display

Figure 6.34

information from the paragraph containing (6.281). Even though the

equation of the ellipse of the figure has already been derived, it is worth-

while to know about the operations involved in using the intrinsic string

property IFiPI + IF2PI = 2a to derive the equation. Letting F,(-ae,0)

and F2(ae,0) be located on the x axis with the origin midway between them

as in Figure 6.34, we use the string property to obtain the uninformative

equation

(6.35) (x + ae)2 + y2 + 1/(x - ae)- + y2 = 2a,

which should be simplified. If we square the members of this equation,

the product of the two square roots will complicate our calculations. It

is better to transpose one of the square roots (we select the second) and

square and simplify the result to obtain

(x-\,I-ex.

Squaring and simplifying this gives

(1 - e2)x2 + y2 = a2(1 - e2),