Answer
(a) See graph and explanations.
(b) vertex $(0,-2)$, directrix $y=-4$.
Work Step by Step
(a) Step 1. 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 $e=1$
Step 2. Examine the equation given in the Exercise $r=\frac{4}{1-sin\theta}$, we can identify that $e=1$ by comparing it with a standard equation in step-1. Thus the graph of the equation will be a parabola.
Step 3. See graph.
(b) With $e=1$ and $ed=4$, we have $d=4$, we can identify that the directrix is $y=-d=-4$. The focus is at the origin and the vertex is at $(0,-2)$ which is half way between the focus and the directrix as indicated in the graph.
