Answer
(a) $2$
(b) Hyperbola
(c) $x=-\frac{5}{4}$
(d) See graph
Work Step by Step
Part (a)
$r=\frac{5}{2-4\cos\theta}$
$\frac{ed}{1-e\cos\theta}=\frac{\frac{5}{2}}{1-2\cos\theta}$
$e=2$ and $ed=\frac{5}{2}$
$e=2$ and $d=\frac{5}{4}$
So, the eccentricity is 2.
Part (b)
Since $e=2>1$, the conic is a hyperbola.
Part (c)
We have obtained $d=\frac{5}{4}$.
Using Theorem 6, and part (b) of Figure 2, the directrix has the equation $x=-\frac{5}{4}$.
Part (d)
We draw the graph.
