Question

Find the equation of hyperbola. Whose axes are coordinate axes and the distance of one of its vertices from the foci are $3$ and $1$.

Answer 1

REF.Image.

Given $B{S}_i=1,A{S}_1=3$

$\Rightarrow ae-a=1$ __ (1) and $2a+ae=3$ __ (2)

solving (1) and (2)

$2a+ae=3$

$2ae-2a=2$

____________

$3ae=5$

$ae=5/3$

and

$2a+ae=3$

$ae-a=1$

____________

$3a=2$

$a=2/3$

Now, we know that

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

$b^2=\frac{25}{9}-\frac{4}{9}=\frac{21}{9}.$

$E{q}^n$ of hyperbola is $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\Rightarrow \frac{9x^2}{4}-\frac{9y^2}{2}=1$