Mathematics - Straight Lines Question with Solution | TestHub

Image of $P$ wrt $x$-axis will be $P^{\prime }(1,-2)$ equation of line joining $P^{\prime }R$ will be $y-3=\frac{5}{3}(x-4)$
Above line will meet $\mathrm{x}$-axis at $\mathrm{Q}$ where $\begin{array}{rl}& \mathrm{y}=0\Rightarrow \mathrm{x}=\frac{11}{5}\\ & \therefore \mathrm{Q}\left(\frac{11}{5},0\right)\end{array}$ $\because \mathrm{PQRS}$ is parallelogram so their diagonals will bisects each other $\begin{array}{rl}& \Rightarrow \frac{4+1}{2}=\frac{\frac{11}{5}+\mathrm{h}}{2}\&\frac{2+3}{2}=\frac{\mathrm{k}+0}{2}\\ & \Rightarrow \mathrm{h}=\frac{14}{5}\&\mathrm{k}=5\\ & \therefore {\mathrm{hk}}^2=\frac{14}{5}\times 5^2=70\end{array}$