Question

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

Answer 1

Let the asymptotes be $x+3y+c_1=0$ and $2x-y+c_2=0$

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

$\therefore ,-2-6+c_1=0$ and $-4+2+c_2=0$

$\Rightarrow c_1=8,c_2=2$

Thus, the equations of the asymptotes are

$x+3y+8=0$ and $2x-y+2=0$

Let the equation of the hyperbola be

$(x+3y+8)(2x-y+2)+\lambda =0$ ........$\left(1\right)$

It passes through $(-1,-4)$.

$\therefore ,(-1-12+8)(-2+4+2)+\lambda =0$

$\Rightarrow -5\times 4+\lambda =0$

$\Rightarrow -20+\lambda =0$

$\Rightarrow \lambda =20$

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

$(x+3y+8)(2x-y+2)+20=0$

This is the equation of the required hyperbola.