Two Arithmetic Families

★ 8/10Arithmetic Progression sequences Trigonometric Identities conic sections analytic geometry

Problem

Let $A, B$ be the left and right vertices of the hyperbola $\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$ ( $a > 0$ , $b > 0$ ), and let $P_1, P_2, \ldots, P_n$ be distinct points on the hyperbola, none of them a vertex. Write $\theta_n = \angle AP_nB$ . If both

\left\{\overrightarrow{P_nA}\cdot\overrightarrow{P_nB}\right\} \qquad \text{and} \qquad \left\{\frac{1}{1 - \cos 2\theta_n}\right\}

are arithmetic sequences with the same common difference, find $\dfrac{1}{a^2} + \dfrac{1}{b^2}$ .

Answer

Solution

Difficulty	8/10
Topics	Arithmetic Progression, sequences, Trigonometric Identities, conic sections, analytic geometry

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