Question

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 $\Delta PQ{Q}^{\prime }$ is

• A. $(6,3)$
• B. $(6,-3)$
• C. $(3,-6)$
• D. $(0,0)$

Answer 1

Answer: (D)

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