Question

The angle of rotation of axes in order to eliminate $xy$ term in the equation $x^2+2\sqrt{3}xy-y^2=2a^2$ is

• A. $\frac{\pi }{6}$
• B. $\frac{\pi }{4}$
• C. $\frac{\pi }{3}$
• D. $\frac{\pi }{2}$

Answer 1

The correct option is A $\frac{\pi }{6}$

Let axes be rotated through an angle $\varphi$, then old co-ordinates are

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

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

$\therefore (x^1\cos \varphi -y^1\sin \varphi {)}^2+2\sqrt{3}(x^1\cos \varphi -y^1\sin \varphi \left)\right(x^1\sin \varphi +y^1\cos \varphi )-(x^1\sin \varphi +y^1\cos \varphi {)}^2=2a^2$

coefficient of $x^1{y}^{1}is$

$-2\cos \varphi \sin \varphi +2\sqrt{3}({\cos }^2\varphi -{\sin }^2\varphi )-2\cos \varphi \sin \varphi =0$

$2\sqrt{3}\left(\cos 2\varphi \right)=2\sin 2\varphi$

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

$\Rightarrow 2\varphi =\frac{\pi }{3}$

$\varphi =\frac{\pi }{6}$

$\therefore$ angle through which it is to be rotated is $\frac{\pi }{6}$