Question

The asymptotes of a hyperbola having centre at the point $(-1,2)$ are parallel to the lines $x-y=0$ and $x+y=0$. If the hyperbola passes through the point $(-3,4)$, Find the equation of the hyperbola.

Answer 1

Let the asymptotes be $x-y+c_1=0$ and $x+y+c_2=0$

Since, the asymptotes passes through the centre $(-1,2)$ of the hyperbola.

$\therefore ,-1-2+c_1=0$ and $-1-2+c_2=0$

$\Rightarrow c_1=3,c_2=3$

Thus, the equations of the asymptotes are

$x-y+3=0$ and $x+y+3=0$

Let the equation of the hyperbola be

$(x-y+3)(x+y+3)+\lambda =0$ ........$\left(1\right)$

It passes through $(-3,4)$.

$\therefore ,(-3-4+3)(-3+4+3)+\lambda =0$

$\Rightarrow -4\times 4+\lambda =0$

$\Rightarrow -16+\lambda =0$

$\Rightarrow \lambda =16$

Putting the value of $\lambda$ in $\left(1\right)$, we obtain

$(x-y-1)(x+y+3)+16=0$

This is the equation of the required hyperbola.