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An ellipse, a lower bound for a ratio

Source: Iran 3rd round 2012-Geometry exam-P3

September 20, 2012
conicsellipseratioanalytic geometrygeometrygeometric transformationdilation

Problem Statement

Cosider ellipse ϵ\epsilon with two foci AA and BB such that the lengths of it's major axis and minor axis are 2a2a and 2b2b respectively. From a point TT outside of the ellipse, we draw two tangent lines TPTP and TQTQ to the ellipse ϵ\epsilon. Prove that TPTQ≥ba.\frac{TP}{TQ}\ge \frac{b}{a}.
Proposed by Morteza Saghafian
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