The asymptotes of the hyperbola xy = hx + ky are

The given hyperbola is 

$xy-hx-lq=0 -----\left(1\right)$

The equation of asymptotes is given by 

$xy-hx-ky+c=0----\left(2\right)$

Equation (2) gives a pair of straight lines. So, 

$\left|\begin{array}{ccc}A& H& G\\ H& B& F\\ G& F& C\end{array}\right|=0$

$\text{or }\left|\begin{array}{ccc}0& 1/2& -\mathrm{h}/2\\ 1/2& 0& -\mathrm{k}/2\\ -\mathrm{h}/2& -\mathrm{k}/2& \mathrm{c}\end{array}\right|=0$

$\begin{array}{l}\mathrm{or} \frac{\mathrm{hk}}{8}+\frac{\mathrm{hk}}{8}-\frac{\mathrm{c}}{4}=0\\ \mathrm{or} \mathrm{c}=\mathrm{hk}\end{array}$

Hence, the asymptotes are 

$\begin{array}{l}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}xy-hx-ky+hk=0\\ \text{ or }\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}(x-k)(y-h)=0\end{array}$