Answer
Ellipse
The directrix is parallel to the polar axis at a distance of 3 units above the pole.
Work Step by Step
We are given the equation in polar coordinates:
$r=\dfrac{6}{8+2\sin\theta}$
Rewrite the equation:
$r=\dfrac{\dfrac{6}{8}}{\dfrac{8+2\sin\theta}{8}}$
$r=\dfrac{\dfrac{3}{4}}{1+\dfrac{1}{4}\sin\theta}$
The equation is in the form:
$r=\dfrac{ep}{1+e\sin\theta}$
Identify $e$ from the denominator, then $p$ from the numerator:
$e=\dfrac{1}{4}$
$ep=\dfrac{3}{4}\Rightarrow \dfrac{1}{4}p=\dfrac{3}{4}\Rightarrow p=3$
Because $e<1$, the conic is an ellipse. The directrix is parallel to the polar axis at a distance of 3 units above the pole.
