Question

lf the axes are rotated through an angle ${30}^0$ about the origin then the transformed equation of $x^2+2\sqrt{3}xy-y^2=2a^2$ is

• A. $X^2+Y^2-a^2=0$
• B. $X^2-Y^2=a^2$
• C. $X^2+Y^2=2a^2$
• D. $X^2-Y^2=2a^2$

Answer 1

Answer: (B)

$X^2-Y^2=a^2$

When axes are rotated through an angle $\theta$, then

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

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

$\therefore x^2+2\sqrt{3}xy-y^2=2a^2$

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

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

$\Rightarrow 2(x^2-y^2)=2a^2[\because {\cos }^2\theta +{\sin }^2\theta =1]$

$x^2-y^2=a^2$