Answer
(a) e = 1/3
(b) Ellipse
(c) x = 9/2
(d) Image below:
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Work Step by Step
(a) $$r = \frac {9}{6 + 2cos(\theta)} \div \frac{6}{6} = \frac {3/2}{1 +\frac 13cos(\theta)}$$ Based on the original equation: $$\frac{ed}{1 + ecos(\theta)}$$ $$e = 1/3$$ (b) $e = 1/3 $, therefore, this equation represents an ellipse.
(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: $c = d$: $$ed = 3/2 \longrightarrow d = \frac{3/2}{e} = \frac{3/2}{1/3} = 9/2 $$ $$x = 9/2$$
(d) Plot the points where $\theta = 0$, $\theta = \pi/2$, $\theta = \pi$ and $\theta = 3\pi/2$
$$r = \frac {9}{6 + 2cos(0)} \longrightarrow (9/8, 0)$$ $$r =\frac {9}{6 + 2cos(\pi/2)} \longrightarrow (3/2, \pi/2)$$ $$r =\frac {9}{6 + 2cos(\pi)}\longrightarrow (9/4, \pi)$$ $$r =\frac {9}{6 + 2cos(3\pi/2)}\longrightarrow (3/2, 3\pi/2)$$ Draw an ellipse that passes through these points.
