Physics

Physics

Problem

A man starts walking from the point P(−3,4)P(-3, 4) touches the xx-axis at RR, and then turns to reach at the point Q(0,2)Q(0, 2). The man is walking at a constant speed. If the man reaches the point QQ in the minimum time, then 50((PR)2+(RQ)2)50\left((PR)^2+(RQ)^2\right) is equal to.
Intuitive explanation

The key insight is that minimum time at constant speed means minimum total distance—so you're looking for the point RR on the xx-axis that minimizes PR+RQPR + RQ. Use the reflection principle: reflect one of the points across the xx-axis, then the shortest path becomes a straight line. Once you find RR, calculate the sum of squared distances and multiply by 50.

Intuitive explanation

The key insight is that minimum time at constant speed means minimum total distance—so you're looking for the point RR on the xx-axis that minimizes PR+RQPR + RQ. Use the reflection principle: reflect one of the points across the xx-axis, then the shortest path becomes a straight line. Once you find RR, calculate the sum of squared distances and multiply by 50.

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