Q. 48 - Calculus

Question

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

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

Step-by-Step Solution

Answer

The graph will be 

Step 1. Given Information.

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

Step 2. Comparison.

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

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

Step 3. Graph.

For $r=\frac{2}{4+\cos \theta }$ the eccentricity is $e=\frac{1}{4}$, so the graph is an ellipse. The directrix will be the vertical line $x=2.$