Answer
Hyperbola (see graph)
Work Step by Step
We are given the polar equation:
$r=\dfrac{8}{4+8\cos\theta}$
Rewrite the equation:
$r=\dfrac{\dfrac{8}{4}}{\dfrac{4+8\cos\theta}{4}}$
$r=\dfrac{2}{1+2\cos\theta}$
The equation is in the form:
$r=\dfrac{ep}{1+e\cos\theta}$
Determine $e$ using the denominator:
$e=2$
Determine $p$ using the numerator:
$ep=2$
$2p=2$
$p=1$
Because e>1, the conic is a hyperbola. Its directrix is perpendicular to the polar axis at a distance of one unit to the right of the pole. The transverse axis is perpendicular to the directrix.
Graph the curve:
