# Chapter 10 - Analytic Geometry - 10.6 Polar Equations of Conics - 10.6 Assess Your Understanding - Page 684: 9

Hyperbola The directrix is parallel to the polar axis at a distance of $\dfrac{4}{3}$ units below the pole.

#### Work Step by Step

We are given the equation in polar coordinates: $r=\dfrac{4}{2-3\sin\theta}$ Rewrite the equation: $r=\dfrac{\dfrac{4}{2}}{\dfrac{2-3\sin\theta}{2}}$ $r=\dfrac{2}{1-\dfrac{3}{2}\sin\theta}$ The equation is in the form: $r=\dfrac{ep}{1-e\sin\theta}$ Identify $e$ from the denominator, then $p$ from the numerator: $e=\dfrac{3}{2}$ $ep=2\Rightarrow \dfrac{3}{2}p=2\Rightarrow p=\dfrac{4}{3}$ Because $e>1$, the conic is a hyperbola. The directrix is parallel to the polar axis at a distance of $\dfrac{4}{3}$ units below the pole.

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