Equation of a Hyperbola Referred to Two Perpendicular Lines

Prove that the straight lines drawn from any point of an equilateral hyperbola to the extremities of any diameter are equally inclined to the asymptotes.

The equations of the transverse and conjugate axes of a hyperbola are respectively $x+2y-3=0, 2x-y+4=0,$ and their respective lengths are $\sqrt{2}$ and $\frac{2}{\sqrt{3}}.$ The equation of the hyperbola is