Question

Find the equation of the hyperbola whose centre is $(1,2)$. The distance between the directrices is $\frac{20}{3}$, the distance between the foci is $30$ and the transverse axis is parallel to $y$-axis.

Answer 1

Since the transerse axis is parallel to $y$-axis, the equation is of the form
$\frac{(y-k{)}^2}{a^2}-\frac{(x-h{)}^2}{b^2}=1$
Here centre $C(h,k)$ is $(1,2)$
The distance between the directrices
$=\frac{2a}{e}=\frac{20}{3}\Rightarrow \frac{a}{e}=\frac{10}{3}$
The distance between the foci,
$2ae=30$

$\Rightarrow ae=15$
$\Rightarrow \frac{a}{e}\left(ae\right)=\frac{10}{3}\times 15$

$\Rightarrow a^2=50$
Also $\frac{ae}{a/e}\Rightarrow e^2=\frac{9}{2}$
$\Rightarrow b^2=a^2(e^2-1)$

$\Rightarrow b^2=50(\frac{9}{2}-1)=175$
This required equation is $\frac{(y-2{)}^2}{50}-\frac{(x-1{)}^2}{175}=1$.