Question

In $\Delta PQR,$ the equation of the internal angle bisector of angle $Q$ is $y=x$ and the equation of side $PR$ is $3x-y=2.$ If coordinates of $P$ are $\left(3,2\right)$ and $2PQ=RQ,$ then the coordinates of $Q$ are

• A. $\left(3,3\right)$
• B. $\left(7,7\right)$
• C. $\left(-2,-2\right)$
• D. $\left(5,5\right)$

Answer 1

Answer: (B)

Point of intersection of $y=x\&3x-y=2$ is $S\left(1,1\right)$
Now, $PS:SR=PR:RQ=1:2$
Hence, using section formula $R=\left(-3,-1\right)$
Let, reflection of P about the line $y=x$ is $P^{{}^{\prime }}$
$\Rightarrow P^{{}^{\prime }}=\left(2,3\right)$ which lies on the side $QR$
$\Rightarrow$ equation of $QR$ is $4x-5y+7=0$
Point of intersection of $QR$ & $y=x$ is
$\left(7,7\right)=Q$