If the point $P(1,3)$ undergoes the following transformations successively. (i) Reflection with respect to the line $y=x$. (ii) Translation through 3 units along the positive direction of the $X$-axis. (iii) Rotation through an angle of $\frac{\pi}{6}$ about the origin in the clockwise direction. Then, the final position of the point $P$ is

• Correct answer
  A

  $\left(\frac{6 \sqrt{3}+1}{2}, \frac{\sqrt{3}-6}{2}\right)$
• B

  $\left(\frac{\sqrt{7}}{2}, \frac{-5}{\sqrt{2}}\right)$
• C

  $\left(\frac{6+\sqrt{3}}{2}, \frac{1-6 \sqrt{3}}{2}\right)$
• D

  $\left(\frac{6+\sqrt{3}-1}{2}, \frac{6+\sqrt{3}}{2}\right)$

Solution

The reflection of the point $P(1,3)$ about the line $y=x$ is $Q(3,1)$
After translation through a distance 3 units along the positive direction of $X$-axis at the point whose coordinate are $R(6,1)$. After rotation through an angle of $\frac{\pi}{6}$ about the origin in the clockwise direction, then $R$ goes to whose coordinates are $$ R^{\prime}\left(\frac{6 \sqrt{3}+1}{2}, \frac{\sqrt{3}-6}{2}\right) $$

Hence, the correct answer is option: 'A'.