Question

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

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

Hint:

Factor the homogeneous equation to find individual lines, then apply rotation.

Answer 1

The Correct Option is C

Step 1: Formula / Definition}
\[ \sqrt{3}x^2 - 4xy + \sqrt{3}y^2 = (\sqrt{3}x - y)(x - \sqrt{3}y) = 0 \]
Step 2: Calculation / Simplification}
Lines: \(y = \sqrt{3}x\) (slope \(\tan 60^\circ\)) and \(y = \frac{1}{\sqrt{3}}x\) (slope \(\tan 30^\circ\))
Rotate by \(30^\circ\) anticlockwise:
\(60^\circ + 30^\circ = 90^\circ \Rightarrow x = 0\)
\(30^\circ + 30^\circ = 60^\circ \Rightarrow y = \sqrt{3}x\)
New pair: \(x(\sqrt{3}x - y) = 0 \Rightarrow \sqrt{3}x^2 - xy = 0\)
Step 3: Final Answer
\[ \sqrt{3}x^2 - xy = 0 \]