Mixed term $xy$ is to be removed from the general equation of second degree $ax^2+2hxy +by^2+2gx+2fy+c=0$,one should rotate the axes through an angle $a$ , then $\tan 2a$ has to be found.
My attempt:
$xy$ term can be zero when $h=0$.
The lines would be real if $a$ and $b$ are of opposite signs.
I tried finding the equation of the two lines but that doesn't seem a correct way to proceed further. Pls help me proceed in a better way...
Thankyou
Expand
$$a(x\cos\alpha-y\sin\alpha)^2+2h(x\cos\alpha-y\sin\alpha)(x\sin\alpha+y\cos\alpha)+b(x\sin\alpha+y\cos\alpha)^2$$ and collect the mixed terms, giving
$$-2a\sin\alpha\cos\alpha+2h(\cos^2\alpha-\sin^2\alpha)+2b\sin\alpha\cos\alpha=0$$
or
$$(b-a)\sin2\alpha+2h\cos2\alpha=0.$$