Question

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

Answer 1

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

Since, the asymptotes passes through the centre $(2,4)$ of the hyperbola.

$\therefore ,4+6+c_1=0$ and $6+8+c_2=0$

$\Rightarrow c_1=-10,c_2=-14$

Thus, the equations of the asymptotes are

$2x+3y–10=0$ and $3x+2y–14=0$

Let the equation of the hyperbola be

$(2x+3y-10)(3x+2y-14)+\lambda =0$ ........$\left(1\right)$

It passes through $(5,3)$.

$\therefore ,(10+9-10)(15+6-14)+\lambda =0$

$\Rightarrow 9\times 7+\lambda =0$

$\Rightarrow \lambda =-63$

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

$(2x+3y–10)(3x+2y-14)–63=0$

This is the equation of the required hyperbola.