A Circle Meets a Hyperbola

★ 7/10Extrema Parametrization conic sections analytic geometry

Problem

Let $F_1, F_2$ be the foci of the hyperbola $C \colon \dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$ ( $a, b > 0$ ), and let $P$ be an intersection point of $C$ with the circle $x^2 + y^2 = r^2$ (as $r$ varies over admissible values). If the maximum value of

\frac{|PF_1| + |PF_2|}{r}

is $4\sqrt{2}$ , find the eccentricity $e$ .

Answer

Solution

Difficulty	7/10
Topics	Extrema, Parametrization, conic sections, analytic geometry

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