Q. 49 - Calculus

Question

Use polar coordinates to graph the conics in Exercises 44–51. 

$r=\frac{2}{4+\sin \theta }$

Step-by-Step Solution

Answer

The graph is

Step 1. Given information.

The given equation is $r=\frac{2}{4+\sin \theta }.$

Step 2. Comparison.

On comparing the given equation with standard equation$r=\frac{eu}{1+e\sin \theta }$,

$$\begin{gathered} r=\frac{2}{4+\sin \theta } \\ r=\frac{\frac{2}{4}}{\frac{4+\sin \theta }{4}} \\ r=\frac{\frac{1}{2}}{\frac{4}{4}+\frac{\sin \theta }{4}} \\ r=\frac{\frac{1}{2}}{1+\frac{1}{4}\sin \theta } \\ \text{ The eccentricity }e=\frac{1}{4} \\ \text{ Here eu }=\frac{1}{2} \\ u=2 \end{gathered}$$

Step 3. Graph.

For $r=\frac{2}{4+\sin \theta }$ the eccentricity is $e=\frac{1}{4}$, so the graph is an ellipse. The directrix is parallel to the polar axis at a distance $u=2$ units above the pole. That means the directrix is $u=2$ units above the pole.