Answer
a. $r= \frac{2}{1+4cos\theta}$
b. $e=4$, $p=\frac{1}{2}$, hyperbola.
c. see figure.
Work Step by Step
a. Rewrite the equation $r=\frac{8}{4+16cos\theta}=\frac{2}{1+4cos\theta}$ as the standard form of a conic in polar coordinates.
b. From the above equation, we can determine $e=4$ and $ep=2$, which gives $p=\frac{1}{2}$. With $e\gt1$, the equation can be identified as a hyperbola.
c. We can graph the polar equation as shown in the figure.
