Question

A straight line through the point $(2,2)$ intersects the lines $\sqrt{3}x+y=0$ and $\sqrt{3}x-y=0$ at the points $A$ and $B$. The equation of the line $AB$, so that the $△OAB$ is equilateral, is

• A. $x-2=0$
• B. $y-2=0$
• C. $x+y-4=0$
• D. $Noneofthese$

Answer 1

The correct option is B $y-2=0$

Given equation of lines are $\sqrt{3}x+y=0$ and $\sqrt{3}x-y=0$
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 ${120}^o$ and ${60}^o$ to 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 $y-2=0$.