Question

The angle through which the coordinates axes be rotated so that \( x y \)-term in the equation: \( 5 x^{2}+4 \sqrt{3} x y+9 y^{2}=0 \) may be missing is - 

(1) \( -\frac{\pi}{6} \)

(2) \( \frac{\pi}{4} \)

(3) \( \frac{\pi}{3} \)

(4) \( \frac{2 \pi}{3} \)

Answer 1

The correct option is (C). \(\frac{\pi}{3}\)

In order to remove xy-term from the equation

ax² + 2hxy + by² + 2gx + 2fy + c, the axes are rotated through an angle θ given by

tan 2θ = \(\frac{2h}{a-b}\)

Here, a = 5, b = 9 and h = 2√3

tan 2θ = \(\frac{2h}{a-b}\)

tan θ = \(\frac{4\sqrt3}{5-9} = -\sqrt3\)

\(2\theta = \frac{2\pi}{3}\)

\(\theta = \frac{\pi}{3}\)