Textbook Problem

Conjecture

Show that the equation of an ellipse can be written as

$\frac{{(x-h)}^2}{a^2}+\frac{{(y-k)}^2}{a^2(1-e^2)}=1$.

(b) Use a graphing utility to graph the ellipse

$\frac{{(x-2)}^2}{4}+\frac{{(y-3)}^2}{4(1-e^2)}=1$ for $e=0.95,e=0.75,e=0.5,e=0.25,\text{and}e=0$

Use the results of part (b) to make a conjecture about the change in the shape of the ellipse as e approaches 0.