MathLabs

Problem 4

Given a plane π\pi, a point PP in π\pi and a point QQ not in π\pi, find all points RR in π\pi such that the ratio QP+PRQR\dfrac{QP+PR}{QR} is a maximum.
Step 2 of 5: Reduce to the ray PX when P≠X
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R∈ray(PX)R\in\text{ray}(PX)
Detailed analysis

Fix PR. On the circle in π centered at P, the numerator is fixed, while QR is smallest when R lies on the line PX on the same side of P as X. Thus every maximizer lies on the ray PX.