Page 76 - Calculus for the Clueless, Calc II

Some of you have seen the equation for an ellipse, but few of you have seen the derivation. It is an excellent
algebraic exercise for you to try. You will see there is a lot that goes into a rather simple equation.

Combine like terms; isolate the
radical.

2 2
2
Reverse sides; take out common factors. Divide on both sides by a (a - c ).

2
2
2
Let a - c = b .

Finally we get

Whew!

We are still not finished. What is a and what is b? Let's investigate.

2
2
2
Since T is any point on the ellipse, F 1T + TF 2 = 2a. By symmetry, F 1T = TF 2. So F 1T= a. Since a - c = b , GT =
b. The coordinates of T are (0,b), and the coordinates of T' are (0,-b).