Question

The pair of lines $\sqrt{3} x^{2}-4 x y+\sqrt{3} y^{2}=0$ are rotated about the origin by $\frac{\pi}{6}$ in the anticlockwise sense. The equation of the pair in the new position is

• A. $x^{2}-\sqrt{3} x y=0$
• B. $x y-\sqrt{3} y^{2}=0$
• C. $\sqrt{3} x^{2}-x y=0$
• D. None of the above

Answer 1

Answer: (C)

The given equation of pair of straight lines can be rewritten as
$(\sqrt{3} x-y)(x-\sqrt{3} y)=0$.
Their separate equations are,
$y=\sqrt{3} x$ and $y=\frac{1}{\sqrt{3}} x$
$\Rightarrow y=\tan 60^{\circ} x$ and $y=\tan 30^{\circ} x$
After rotation, the separate equations are
$y=\tan 90^{\circ} x$ and $y=\tan 60^{\circ} x$
$\Rightarrow x=0$ and $y=\sqrt{3} \cdot x$
$\therefore $ Combined equation in the new position is
$x(\sqrt{3} x-y)=0$ or $\sqrt{3} x^{2}-x y=0$