Question

The asymptotes of a hyperbola having centre at the point $(-1,-4)$ are parallel to the lines $2x-3y=0$ and $x+5y=0$. If the hyperbola passes through the point $(-2,-3)$, Find the equation of the hyperbola.

Answer 1

Let the asymptotes be $2x-3y+c_1=0$ and $x+5y+c_2=0$

Since, the asymptotes passes through the centre $(-1,-4)$ of the hyperbola.

$\therefore ,-2+12+c_1=0$ and $-1-20+c_2=0$

$\Rightarrow c_1=-10,c_2=21$

Thus, the equations of the asymptotes are

$2x-3y-10=0$ and $x+5y+21=0$

Let the equation of the hyperbola be

$(2x-3y-10)(x+5y+21)+\lambda =0$ ........$\left(1\right)$

It passes through $(-2,-3)$.

$\therefore ,(-4+9-10)(-2-15+21)+\lambda =0$

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

$\Rightarrow -20+\lambda =0$

$\Rightarrow \lambda =20$

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

$(2x-3y-10)(x+5y+21)+20=0$

This is the equation of the required hyperbola.