the equation $x^2-y^2 =a^2$ changes to the form $xy=c^2$ if the co-ordinate axes rotates through an angle (keeping origin fixed)

Question

the equation $x^2-y^2 =a^2$ changes to the form $xy=c^2$ if the co-ordinate axes rotates through an angle

(keeping origins fixed) is

a) $ \frac \pi 2 $ b) $ - \frac \pi 2 $ c) $ \frac \pi 4 $ d) $ \frac \pi 3 $

Answer 1

If we put $$x=\frac{X+Y}{\sqrt{2}}$$ and $$y=\frac{X-Y}{\sqrt{2}}$$

then

$$x^2-y^2=a^2=2XY$$ or $$XY=(\frac{a}{\sqrt{2}})^2=c^2$$

But $$\cos(\frac{\pi}{4})=\frac{1}{\sqrt{2}}$$ the right answer is $ c)$.