P is a point on either of the two lines $y-\sqrt{3}\left|x\right|=2$ at a distance of 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

Equation, of two lines are $y=\sqrt{3}x+2, if x\ge 0$  and $y=-\sqrt{3}x+2, if x\ge 0$

Clearly, $y\ge 2$

Also, y-axis is the bisector of the angle between the two lines. $P_1,P_2$ are two points on these lines, at a distance 5 units from A.. Q is the foot of the $\perp$ from $P_1$ and $P_2$ on the bisector (y-axis). 

Then, the coordinates of Q are (0, 2 + 5 cos 30º)

$=\left(0,2+\frac{5\sqrt{3}}{2}\right)=\left(0,\frac{1}{2}\left(4+5\sqrt{3}\right)\right)$