Answer
(a) $e=1$, directrix $x=\frac{9}{2}$.
(b) $r=\frac{9}{2+2cos(\theta+\frac{5\pi}{6})}$
(c) See graph.
Work Step by Step
The standard form of a conic polar equation is $r=\frac{ed}{1\pm e\cdot cos\theta}$ or $r=\frac{ed}{1\pm e\cdot sin\theta}$. The graph of the equation will be a parabola if $e=1$, an ellipse if $e\lt1$, and a hyperbola if $e\gt1$
(a) Rewrite the equation given in the Exercise as $r=\frac{9/2}{1+cos\theta}$, we can identify that $e=1$ by comparing it with a standard equation above. Thus the graph of the equation will be a parabola. With $ed=\frac{9}{2}$, we get $d=\frac{9}{2}$ and the directrix is given by $x=d=\frac{9}{2}$.
(b) The equation after the rotation can be written as $r=\frac{9}{2+2cos(\theta+\frac{5\pi}{6})}$
(c) See graph.
