One asymptotes of the rectangular hyperbola is $x+2y-5=0$.The other asymptote is $\perp$ to it.The equation is of the form $2x-y+l=0.$
The combined equation is $(x+2y-5)(2x-y+l)=0$
The equation of the hyperbola differs only in the constant term.
Therefore its equation is of the form $(x+2y-5)(2x-y+l)=k$------(1)
The rectangular hyperbola passes through $(6,0)$ and $(-3,0)$.
Substituting these in equ(1) we get
Solving eq(2) & eq(3) we get
Substitute the value of $l$ in eq(2) we get
The equation of the rectangular hyperbola is $ (x+2y-8)(2x-y+4)=16$
On simplifying we get,
This is the required equation.