Page 103 - Calc for the Clueless

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.

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
2
2
2
= b. The coordinates of T are (0,b), and the coordinates of T' are (0,-b).

We would like to find the coordinates of U and we have used up the letters a, b, and c. Oh well, let's see what
happens. F 2U + UF 1 = 2a. F 2U = x - c. UF 1 = x + c. x + c + x - c = 2a. So x = a. The coordinates of U are (a,0).
The coordinates of U' are (-a,0).

c = half the distance between the foci. b = length of the semiminor axis ("semi" means half, "minor" means
smaller, "axis" means line), a = length of the semimajor axis = distance from a focus to a minor vertex. (±a,0)
are the major vertices. (0,±b) are the minor vertices or co-vertices. (±c,0) are the foci.

Although the derivation is very long, sketching should be short.

Example 16—