Question

Axes are rotated through $a+ive$ obtuse angle $\theta$ so that the transformed equation of the curve $3x^2-6xy+3y^2+7x-3=0$ is free from the term of $xy$ then the coefficient of $x^2$ in the transformed equation is ____

Answer 1

$6$.
$3(xcos\theta -ysin\theta {)}^2-6(xcos\theta -ysin\theta )$
$(xsin\theta +ycos\theta )+3(xsin\theta +ycos\theta {)}^2+7(xcos\theta -ysin\theta )-3=0$
Coefficient of $xy=0$
$⟹-6sin\theta cos\theta -6(co{s}^2\theta -si{n}^2\theta )+6sin\theta cos\theta =0$
$thereforecos2\theta =0$ $\therefore =\frac{\pi }{2},\frac{3\pi }{2}$ $\therefore \theta =\frac{\pi }{4},\frac{3\pi }{4}$
Since $\theta$ is obtuse we choose $\theta =\frac{3\pi }{4}$
$\therefore$ Coefficient of $x^2$
$=3(co{s}^2\theta +si{n}^2\theta )-6sin\theta cos\theta$
$=3.1-3.sin2theta=3+3=6$ $\because sin2\theta =sin\frac{3\pi }{2}=-1$