Question

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

Answer 1

Let the asymptotes be $3x+4y+c_1=0$ and $4x+5y+c_2=0$

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

$\therefore ,3+8+c_1=0$ and $4+10+c_2=0$

$\Rightarrow c_1=-11,c_2=-14$

Thus, the equations of the asymptotes are

$3x+4y-11=0$ and $4x+5y-14=0$

Let the equation of the hyperbola be

$(3x+4y-11)(4x+5y-14)+\lambda =0$ ........$\left(1\right)$

It passes through $(3,5)$.

$\therefore ,(9+20-11)(12+25-14)+\lambda =0$

$\Rightarrow 18\times 23+\lambda =0$

$\Rightarrow 414‬+\lambda =0$

$\Rightarrow \lambda =-414‬$

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

$(3x+4y-11)(4x+5y-14)-414=0$

This is the equation of the required hyperbola.