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  4. Let the images of the point A(2,3) about the lines y=x and y=mx are P and Q respectively. If the line PQ passes through the origin, then m is equal to

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Q. Let the images of the point $A\left(2,3\right)$ about the lines $y=x$ and $y=mx$ are $P$ and $Q$ respectively. If the line $PQ$ passes through the origin, then $m$ is equal to

NTA AbhyasNTA Abhyas 2022

Solution:

Solution
Reflection of $A\left(2,3\right)$ about line $y=x$ is $\left(3,2\right)=P$
Equation of the line $PQ$ is the same as line $OP$
$\Rightarrow $ equation of the line $PQ$ is $y=\frac{2}{3}x$ and the equation of the line $OA$ is $y=\frac{3}{2}x$ .
Now, $y=mx\&y=x$ are the angle bisectors of the lines $2x-3y=0$ and $3x-2y=0$
Now, angle bisectors of $2x-3y=0$ and $3x-2y=0$ are $y=x$ and $y=-x$
$\Rightarrow m=-1$