A Dot Product Fixes $e$

★ 7/10Midpoint Chord Method conic sections analytic geometry vectors

Problem

Let $O$ be the origin, and let $A, B, C$ be three points on the ellipse $E \colon \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ (with $a > b > 0$ ) satisfying

\overrightarrow{OA} + \overrightarrow{OB} = \overrightarrow{0}, \qquad \overrightarrow{OA}\cdot\overrightarrow{AC} = 0.

The line $BC$ meets the $x$ -axis at $D$ , and

4\,\overrightarrow{OA}\cdot\overrightarrow{OD} = \overrightarrow{OD}^2.

Find the eccentricity $e$ of $E$ .

Answer

Solution

Difficulty	7/10
Topics	Midpoint Chord Method, conic sections, analytic geometry, vectors

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