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2006 China Second Round Olympiad Test 1 #9

Source:

September 28, 2014
conicsellipseratiogeometry

Problem Statement

Suppose points F1,F2F_1, F_2 are the left and right foci of the ellipse x216+y24=1\frac{x^2}{16}+\frac{y^2}{4}=1 respectively, and point PP is on line l:l:, x−3y+8+23=0x-\sqrt{3} y+8+2\sqrt{3}=0. Find the value of ratio ∣PF1∣∣PF2∣\frac{|PF_1|}{|PF_2|} when ∠F1PF2\angle F_1PF_2 reaches its maximum value.
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