The conbined equation of the diagonals of the square formed by the pairs of lines $xy+6y-4x-24=0$ and $xy+6x-4y-24=0$ is

$$\begin{gathered} xy+6y-4x-24=0 \\ \begin{array}{l}(x+6)(y-4)=0\\ x+6=0andy-4=0\mathrm{....}\left(1\right)\\ andxy+6x-4y-24=0\\ (x-4)(y+6)=0\\ x-4=0andy+6=0\end{array} \end{gathered}$$
Now eq of diagonal AC is $y-4=1(x-4)$
$x-y=0$
And equation BD is $y-4=-1(x+6)$
$$\begin{gathered} \Rightarrow x+y+2=0 \\ \begin{array}{l}(x-y)(x+y+2)=0\\ x^2-y^2+2x-2y=0\end{array} \end{gathered}$$