CAT 2015 Quantitative Ability

PQ is perpendicular to line y = $\frac{\mathit{x}}{\sqrt{3}}$
∴ Slop of PQ = $\frac{\frac{-1}{1}}{\sqrt{3}}=-\sqrt{3}$
∴ Let equation of line PQ be y =$-\sqrt{3}\mathit{x}+\mathit{c}$
At point M, when $\mathit{x}=\sqrt{3.}\mathit{y}=1$
∴ c = 4
⇒ $\mathit{y}=-\sqrt{3}\mathit{x}+4$
∴ Co-ordinates of point Q =$\left(\frac{4}{\sqrt{3}},0\right)$ and Co-ordinates of point P = (0,4).
Hence, PQ = $\sqrt{{\left(\frac{4}{\sqrt{3}}\right)}^2}+4^2=4\sqrt{\frac{1}{3}+1}$