A Harmonic Division

★ 7/10Pole and Polar Parametrization conic sections analytic geometry

Problem

The hyperbola $C \colon \dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$ (with $a > 0$ , $b > 0$ ) passes through the point $A\left(4\sqrt{2}, 3\right)$ and has focal distance $10$ .

1. Find the equation of $C$ . 2. Let $B\left(4\sqrt{2}, -3\right)$ and $D\left(2\sqrt{2}, 0\right)$ . Let $E$ be a point on segment $AB$ such that the line $DE$ meets $C$ at two points $G$ and $H$ . Prove that

\frac{|GD|}{|GE|} = \frac{|HD|}{|HE|}.

Answer

Solution

Difficulty	7/10
Topics	Pole and Polar, Parametrization, conic sections, analytic geometry

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