Answer
(a) e = 2
(b) Hyperbola
(c) x = -3/8
(d) Image below:
Work Step by Step
(a) $$r = \frac {3}{4 - 8cos(\theta)} \div \frac{4}{4} = \frac {3/4}{1 - 2cos(\theta)}$$ Based on the original equation: $$\frac{ed}{1 - ecos(\theta)}$$ $$e = 2$$ (b) $e = 2 $, therefore, this equation represents a hyperbola.
(c) The equation has a $-cos(\theta)$ and the focus is at (0,0), thus, the equation of the directrix must be in the "$x = -c$" pattern. In this case: $c = d$: $$ed = 3/4 \longrightarrow d = \frac{3/4}{e} = \frac{3/4}{2} = 3/8 $$ $$x = -3/8$$
(d) Plot the points where $\theta = 0$, $\theta = \pi/2$, $\theta = \pi$ and $\theta = 3\pi/2$
$$r = \frac {3}{4 - 8cos(0)} \longrightarrow (-3/4, 0)$$ $$r = \frac {3}{4 - 8cos(\pi/2)} \longrightarrow (3/4, \pi/2)$$ $$r = \frac {3}{4 - 8cos(\pi)}\longrightarrow (1/4, \pi)$$ $$r = \frac {3}{4 - 8cos(3\pi/2)}\longrightarrow (3/4, 3\pi/2)$$
Draw a hyperbola that passes through these points.
