A Reciprocal Slope Relation

★ 9/10Substitution conic sections analytic geometry Vieta's Formulas

Problem

Consider the hyperbola $\dfrac{x^2}3-y^2=1$ with left and right foci $F_1$ , $F_2$ . Let $P$ be a moving point on the hyperbola. The lines $PF_1$ and $PF_2$ meet the hyperbola again at points $M$ and $N$ respectively. Let $k_1,k_2,k_3,k_4$ be the slopes of $MF_2$ , $NF_1$ , $PF_1$ , $PF_2$ . Prove that there is a constant $\mu$ with

\frac 1{k_1}+\frac 1{k_2}=\mu\left(\frac 1{k_3}+\frac 1{k_4}\right).

Answer

Solution

Difficulty	9/10
Topics	Substitution, conic sections, analytic geometry, Vieta's Formulas

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