The equation of the transverse and conjugate axis of a hyperbola are $x+2y-3=0$ and $2x-y+4=0$, and their corresponding lengths are $\sqrt{2}$ and $\dfrac{2}{\sqrt{3}}$. Find the equation of the hyperbola:
This is attempted to solve in my reference as
$$ \frac{\big(\frac{x+2y-3}{\sqrt{5}}\big)^2}{\frac{1}{2}}-\frac{\big(\frac{2x-y+4}{\sqrt{5}}\big)^2}{\frac{1}{3}}=1\\ \frac{2(x+2y-3)^2}{5}-\frac{3(2x-y+4)^2}{5}=1 $$ I am having trouble understanding the logic behind such a substitution in the above attempt ?