Question

If extremities of diameter of the circle $x^{2}+y^{2}=16$ are foci of an ellipse, then find the eccentricity of the ellipse, if its size is just sufficient to contain the circle.

Answer 1

$2 a e=8$
$b=4$
$b^{2}=a^{2}-a^{2} e^{2}$
$16=a^{2}-16$
$a^{2}=32$
$e=\sqrt{1-\frac{16}{32}}=\frac{1}{\sqrt{2}}$