An Ellipse from a Combination

★ 7/10Substitution conic sections analytic geometry Vieta's Formulas vectors

Problem

The ellipse $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ (with $a > b > 0$ ) has eccentricity $\dfrac{\sqrt{3}}{2}$ . The line $y = \dfrac{1}{2}x + 1$ meets the ellipse at $A$ and $B$ , and a point $M$ on the ellipse satisfies

\overrightarrow{OM} = \frac{1}{2}\overrightarrow{OA} + \frac{\sqrt{3}}{2}\overrightarrow{OB}.

Find the equation of the ellipse.

Answer

Solution

Difficulty	7/10
Topics	Substitution, conic sections, analytic geometry, Vieta's Formulas, vectors

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