Page 77 - Calculus for the Clueless, Calc II

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).

Now, 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,O) 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 6—

Sketch x /11 + y /8 = 1.
2
2
In the case of an ellipse, the longer axis is indicated by which number is larger under x or y . That term is a .
2
2
2
(Try not to remember a or b—remember the figure.) This ellipse is longer in the x direction.
Letting y = 0, we get the major vertices . Letting x = 0, we get the minor vertices .
. The foci are . See the figure.