The point $P(3,6)$ is first reflected on the line $y=x$ and then the image point $Q$ is again reflected on the line $y=-x$ to get the image point $Q^{\prime} .$ Then, the circumcentre of the $\Delta P Q Q^{\prime}$ is

• A

  (6,3)
• B

  (6,-3)
• C

  (3,-6)
• Correct answer
  D

  (0,0)

Solution

The coordinates of $Q$ and $Q^{\prime}$ will be (6,3) and $(-3,-6),$ respectively. Now, slope of $P Q=\frac{6-3}{3-6}=-1$ and Slope of $Q Q^{\prime}=\frac{-6-3}{-3-6}=1$ $\therefore$ Slope of $P Q \times$ Slope of $Q Q^{\prime}=-1 \times 1=-1$ $\therefore \Delta P Q Q^{\prime}$ is right angled triangle at $Q .$ $\therefore$ Circumcentre will be the mid-point of hypotenuse $P Q^{\prime}=\left(\frac{-3+3}{2}, \frac{-6+6}{2}\right)$ $$ =(0,0) $$

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