2020-II-24

Ans: D
Since the coordinates of the reflection image of $Q$ with respect to the $y$-axis are $(5,-1)$, then the coordinates of $Q$ are $(-5,-1)$.

Since $P$ is translated leftwards by $4$ units to $Q$, then the coordinates of $P$ are $(-1,-1)$.

By Pythagoras Theorem, we have

$\begin{array}{rcl}O{P}^2& =& 1^2+1^2\\ O{P}^2& =& 2\\ OP& =& \sqrt{2}\end{array}$

Also, we have

$\begin{array}{rcl}\tan \theta & =& {\displaystyle \frac{1}{1}}\\ \tan \theta & =& 1\\ \theta & =& {45}^{\circ }\end{array}$

Therefore, the polar coordinates of $P$ are $(\sqrt{2},{45}^{\circ }+{180}^{\circ })$, i.e. $(\sqrt{2},{225}^{\circ })$.