Question

lf the equation $4x^2+2\sqrt{3}xy+2y^2-1=0$ becomes $5X^2+Y^2=1$, when the axes are rotated through an angle $\theta$, then $\theta$ is

• A. ${15}^o$
• B. ${30}^o$
• C. ${45}^0$
• D. ${60}^o$

Answer 1

The correct option is B ${30}^o$

By rotation of axes through $\theta$, co-ordinates become

$x=x^1\cos \theta -y^1\sin \theta$

$y=x^1\sin \theta +y^1\cos \theta$

$4x^2+2\sqrt{3}xy+2y^2-1=0$

$\Rightarrow 4(x^1\cos \theta -y^1\sin \theta {)}^2+2\sqrt{3}(x^1\cos \theta -y^1\sin \theta \left)\right(x^1\sin \theta +y^1\cos \theta )+2(x^1\sin \theta +y^1\cos \theta {)}^2-1=0$

coeff of $xy=0$

$\Rightarrow 4(-\sin 2\theta )+2\sqrt{3}\cos 20+2\sin 2\theta =0$

$2\sqrt{3}\cos 2\theta =2\sin 2\theta$

$\tan 2\theta =\sqrt{3}$

$2\theta =\frac{\pi }{3}$

$\theta =\frac{\pi }{6}={30}^{\circ }$

$\therefore$ angle to be rotated $={30}^{\circ }$