If M is a point on the line y=x and points P( | vedclass

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(A) Let the point $M$ be $(a, a)$ since it lies on the line $y=x$. To minimize the sum of distances $PM + QM$,where $P(0,1)$ and $Q(2,0)$ are on the s

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$(0,0)$

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(A) Let the point $M$ be $(a, a)$ since it lies on the line $y=x$. To minimize the sum of distances $PM + QM$,where $P(0,1)$ and $Q(2,0)$ are on the same side of the line $y=x$,we reflect point $P$ across the line $y=x$ to get $P'(1,0)$. The minimum distance $PM + QM$ is equal to the distance $P'Q$ when $P', M, Q$ are collinear. The line passing through $P'(1,0)$ and $Q(2,0)$ is the $x$-axis $(y=0)$. However,$M$ must lie on $y=x$. The intersection of $y=x$ and $y=0$ is $(0,0)$. Wait,let us re-evaluate using the reflection principle: Reflect $P(0,1)$ across $y=x$ to get $P'(1,0)$. The distance $PM + QM = P'M + QM$. This is minimized when $P', M, Q$ are collinear. The line $P'Q$ passes through $(1,0)$ and $(2,0)$,which is the line $y=0$. The intersection of $y=x$ and $y=0$ is $(0,0)$. Checking the options,the provided solution logic in the prompt used the slope method incorrectly for points on the same side. Actually,for $P(0,1)$ and $Q(2,0)$,the reflection of $P$ across $y=x$ is $P'(1,0)$. The line $P'Q$ is $y=0$. The intersection of $y=x$ and $y=0$ is $(0,0)$. Given the options,let's re-verify the reflection of $Q(2,0)$ across $y=x$ which is $Q'(0,2)$. The line $PQ'$ passes through $(0,1)$ and $(0,2)$,which is $x=0$. The intersection of $x=0$ and $y=x$ is $(0,0)$. Thus,the minimum occurs at $(0,0)$.

If $M$ is a point on the line $y=x$ and points $P(0,1), Q(2,0)$ are such that $PM+QM$ is minimum,the coordinates of $M$ are

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