Answer
(a) $e=\frac{3}{4}$, directrix $x=-\frac{1}{3}$.
(b) $r=\frac{1}{4-3cos(\theta-\frac{\pi}{3})}$
(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 $r=\frac{1/4}{1-\frac{3}{4}cos\theta}$, we can identify that $e=\frac{3}{4}\lt1$ by comparing it with a standard equation above. Thus the graph of the equation will be an ellipse. With $ed=\frac{1}{4}$, we get $d=\frac{1}{3}$ and the directrix is given by $x=-d=-\frac{1}{3}$.
(b) The equation after the rotation can be written as $r=\frac{1}{4-3cos(\theta-\frac{\pi}{3})}$
(c) See graph.
