Question

$P$ is a point on either of the two lines $y-\sqrt{3}\left|x\right|=2$ at a distance $5$ units from their point of intersection. The coordinates of the foot of the perpendicular from $P$ on the bisector of the angle between them are

• A. $(1,\frac{4+5\sqrt{3}}{2})$ or $(-1,\frac{4-5\sqrt{3}}{2})$depending on which the point $P$ is taken
• B. $(0,\frac{4+5\sqrt{3}}{2})$
• C. $(0,\frac{4-5\sqrt{3}}{2})$
• D. $\left(\frac{5}{2},\frac{5\sqrt{3}}{2}\right)$

Answer 1

The correct option is B $(0,\frac{4+5\sqrt{3}}{2})$

$y-\sqrt{3}x=2$ ---(1) and

$y+\sqrt{3}x=2$ ---(2)
From $1$ & $2,$ $y=2$ and $x=0$

$y-\sqrt{3}|x|=2,y>2$

The $y-$ axis is the angle bisector of the two given lines.

Let us find the coordinates of $P$ by using the parametric form of the line.
$P\equiv (5\sin {30}^{\circ },2+5\cos {30}^{\circ })$ or $P\equiv (-5\sin {30}^{\circ },2+5\cos {30}^{\circ })$
$P\equiv \left(\frac{\pm 5}{2},2+\frac{5\sqrt{3}}{2}\right)$
To find the foot of the perpendicular on the $y-$ axis, we will draw a line parallel to the $x-$ axis.

Hence, the coordinates of the foot of the perpendicular will be $(0,\frac{4+5\sqrt{3}}{2})$