Question

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

Answer 1

Let the equation of the ellipse be
$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ ...(i)
We have,
2ae=8 [given]
$\Rightarrow e=\frac{8}{2a}$
$\Rightarrow e=\frac{4}{a}$ ...(ii)
Now,
$\frac{2a}{e}=18$ [given]
$\Rightarrow a=\frac{18e}{2}$
$\Rightarrow a=9e$ ...(iii)
Using equation (ii) and equation (iii), we get
$a=\frac{9\times 4}{a}$
$\Rightarrow a^2=36$
Now,
$b^2=a^2(1-e^2)$
$\Rightarrow b^2=36-(ae{)}^2$
$\Rightarrow b^2=36-16$
$\Rightarrow b^2=20$
Putting $a^2=36$ in $b^2=20$ in equation (i), we get
$\frac{x^2}{36}+\frac{y^2}{20}=1$
This is the equation of the required ellipse.