Question

Find the equation of an ellipse, the distance between the foci is 8 units and the distance between the directrices is 18 units.

• A. $\frac{x^2}{25}+\frac{y^2}{36}=1$
• B. $\frac{x^2}{36}+\frac{y^2}{20}=1$
• C. $\frac{x^2}{20}+\frac{y^2}{25}=1$
• D. None of these

Answer 1

Answer: (B)

$\frac{x^2}{36}+\frac{y^2}{20}=1$

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

We have,

$2ae=8$ and $\frac{2a}{e}=18$

$2ae\times \frac{2a}{e}=8\times 18$

$4a^2=8\times 18$

$a=6$

Now, $2ae=8$ and $a=6⟹e=\frac{2}{3}$

Therefore, $b^2=a^2(1-e^2)$

$b^2=36(1-\frac{4}{9})=20$

Hence, the equation of the ellipse is $\frac{x^2}{36}+\frac{y^2}{20}=1$