Answer
(a) $e=1$, directrix $y=2$.
(b) $r=\frac{2}{1+sin(\theta+\frac{\pi}{4})}$
(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) Write the equation given in the Exercise as $r=\frac{2}{1+sin\theta}$, 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 $ed=2$, we get $d=2$ and the directrix is given by $y=d=2$.
(b) The equation after the rotation can be written as $r=\frac{2}{1+sin(\theta+\frac{\pi}{4})}$
(c) See graph.
