Find the equation of the ellipse in the standard form such that distance between foci is 8 and distance between directrices is 32.

Let the ellipse be$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$

Distance b/w foci is ‘8’
$2\mathrm{ae}=8\Rightarrow \mathrm{ae}=4 ...\left(1\right)$
And distance b/w directrices is 32
$\begin{array}{l}\frac{2\mathrm{a}}{\mathrm{e}}=32\Rightarrow \frac{\mathrm{a}}{\mathrm{e}}=16\dots \dots \left(2\right)\\ \left(1\right)\times \left(2\right) \Rightarrow \mathrm{ae}\cdot \frac{\mathrm{a}}{\mathrm{e}}=4\times 16\\ \Rightarrow {\mathrm{a}}^2=64\end{array}$
Now, ${\mathrm{b}}^2={\mathrm{a}}^2\left(1-{\mathrm{e}}^2\right)={\mathrm{a}}^2-{\mathrm{a}}^2{\mathrm{e}}^2$
= 64 – 16 = 48
$\therefore$Equation of ellipse is $\frac{{\mathrm{x}}^2}{64}+\frac{{\mathrm{y}}^2}{48}=1$