Question

In an ellipse, the distance between its foci is $6$ and minor axis is $8$. If its eccentricity is $\frac{p}{q}$ then $p + q$ is

Answer 1

$2ae = 6 \Rightarrow ae = 3; 2b =8 \Rightarrow b =4$
$b^{2} =a^{2} \left(1 -e^{2}\right) ;16 =a^{2} -a^{2} e^{2}$
$\Rightarrow a^{2} = 16 +9 =25$
$\Rightarrow a =5$
$\therefore e =\frac{3}{a} =\frac{3}{5}$
$\Rightarrow \frac{p}{q} =\frac{3}{5}$
$\Rightarrow p =3, q =5$
Hence, $p +q = 3 +5 =8$