## Precalculus (10th Edition)

Published by Pearson

# Chapter 10 - Analytic Geometry - 10.6 Polar Equations of Conics - 10.6 Assess Your Understanding - Page 684: 11

#### Answer

Ellipse The directrix is perpendicular to the polar axis at a distance of $\dfrac{3}{2}$ units to the left of the pole.

#### Work Step by Step

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

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