Q. A straight line through the point (2, 2) intersects the lines $ \sqrt{3}x+y=0 $ and $ \sqrt{3x}-y=0 $ at the points A and B. The equation of the line AB, so that the $ \Delta \text{OAB} $ is equilateral, is
Bihar CECEBihar CECE 2014
Solution:
Given equations of lines are $ \sqrt{3x}+y=0 $ and $ \sqrt{3x}-y=0 $
The slopes of the lines are $ \tan {{\theta }_{1}}=-\sqrt{3} $ and $ \tan {{\theta }_{2}}=\sqrt{3} $ $ \Rightarrow $ $ {{\theta }_{1}}={{120}^{o}} $ and $ {{\theta }_{2}}={{60}^{o}} $ Thus, the lines make angles $ \text{12}0{}^\circ $ and $ \text{6}0{}^\circ $ 1o the, X-axis. Any line parallel to X-axis forms an equilateral triangle and it passes through the point (2,2). Hence, equation of required line is y=2 or y-2=0,
