question_answer

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       

A) x-2=0                                    

B) \[y-2=0\]

C) x + y - 4 = 0         

D) None of these

Correct Answer: B

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,