Question

Find the asymptotes of $xy-3y-2x=0.$

Answer 1

Since equation of a hyperbola and its asymptotes differ in constant terms only,

$\therefore$ Pair of asymptotes is given by $xy-3y-2x+\lambda =0$

where $\lambda$ is any constant such that it represents two straight lines.

$\therefore abc+2fgh-a{f}^2-b{g}^2-c{h}^2=0\Rightarrow 0+2\times -\frac{3}{2}\times -1\frac{1}{2}-0-0-\lambda {\left(\frac{1}{2}\right)}^2=0\therefore \lambda =6$
From (1) , the asymptotes of given hyperbola are given by

$xy-3y-2x+6=0or(y-2)(x-3)=0\therefore$ Asymptotes are $x-3=0$ and $y-2=0$