Question

If $2\cos \theta -\sin \theta =x$ & $\cos \theta -3\sin \theta =y$, Prove that $2x^2+y^2-2xy=5$.

Answer 1

To prove $2x^2+y^2-2xy=5$

i.e, $x^2+x^2+y^2-2xy=5$

$\therefore x^2+(x-y{)}^2=5$

$\therefore x=2\cos \theta -\sin \theta$

$\therefore x^2+4{\cos }^2\theta +{\sin }^2\theta -4\sin \theta \cos \theta ....\left(1\right)$

$x-y=\cos \theta +2\sin \theta$

$\therefore (x-y{)}^2={\cos }^2\theta +4{\sin }^2\theta +4\cos \theta \sin \theta ......(2)$

Adding $\left(1\right)$ & $\left(2\right)$

$5({\sin }^2\theta +{\cos }^2\theta )=5$