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 $△PQ{Q}^{\prime }$ is:

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

Answer 1

The correct option is D $(0,0)$

$P(3,6)$
Point P is reflected on line $y=x$
$\therefore$ Now point $Q(6,3)$ [interchange x & y ]
Q is reflected on line $y=-x$
$\therefore$ Interchange x & y of Q point and put negative sign for y
$\therefore Q^{{}^{\prime }}(3,-6)$
Let $O(x,y)$ be circumcentre of $△PQ{Q}^{{}^{\prime }}$
Distance $OP=OQ=O{Q}^{{}^{\prime }}$
$OP=OQ$
${(x-3)}^2+{(y-6)}^2={(x-6)}^2+{(y-3)}^2$
$x^2+9-6x+y^2-12y+36=x^2-12x+36+y^2+9-6y$
$6x-6y=0$
$x=y$
$OP=O{Q}^{{}^{\prime }}$
${(x-3)}^2+{(y-6)}^2={(x-3)}^2+{(y+6)}^2$
$y^2+36-12y=y^2+36+12y$
$24y=0$
$y=0$
$\therefore x=0$
$\therefore (0,0)$ is circumcentre