A Three-to-One Focal Ratio

★ 6/10Substitution conic sections analytic geometry vectors

Problem

Let $F$ be a focus of the hyperbola $C \colon \dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$ (with $a, b > 0$ ). A line through $F$ is perpendicular to one asymptote, meeting it at $A$ , and meets the other asymptote at $B$ . If

3\overrightarrow{AF} + \overrightarrow{BF} = \overrightarrow{0},

find the eccentricity of the hyperbola $C$ .

Answer

Solution

Difficulty	6/10
Topics	Substitution, conic sections, analytic geometry, vectors

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