Question

If a straight line parallel to the line $y=\sqrt{3}x$ passes through $Q(2,3)$ and cuts the line $2x+4y-27=0$ at $P$, then the length of $PQ$ is (units)

• A. $2\sqrt{3}-1$
• B. $2\sqrt{3}+1$
• C. $3\sqrt{3}-1$
• D. $3\sqrt{3}+1$

Answer 1

The correct option is A $2\sqrt{3}-1$

Since $PQ$ is parallel to the straight line $y=\sqrt{3}x$.
$\therefore \text{slope}=\sqrt{3}$
The line makes an angle ${60}^{\circ }$ with positive direction of $\text{x-axis}$
Coordinates of $P$ are $(2+r\cos {60}^{\circ },3+r\sin {60}^{\circ })$
i.e.,$(2+\frac{r}{2},3+\frac{\sqrt{3}r}{2}),\text{where}|r|=PQ$
$\because P$ lie on the line $2x+4y-27=0$
$\therefore 2(2+\frac{r}{2})+4(3+\frac{\sqrt{3}r}{2})=27\Rightarrow r=2\sqrt{3}-1\therefore PQ=|r|=2\sqrt{3}-1\text{units}$