Question

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

$r=\frac{2}{1+3\cos \theta }$

Short answer

The graph is

Step-by-step solution

Step 1. Given information.

$r=\frac{2}{1+3\cos \theta }.$The given equation is

Step 2. Comparison.

On comparing the given equation with the standard form,

$$\begin{gathered} r=\frac{eu}{1+e\cos \theta } \\ \text{The eccentricity}e=3 \\ \text{Here}eu=2 \\ 3·u=2\Rightarrow u=\frac{2}{3} \end{gathered}$$

Step 3. Graph of the equation.

For $r=\frac{2}{1+3\cos \theta }$the eccentricity is $e=2$, so the graph is a hyperbola. The directrix is perpendicular to the polar axis at a distance $u=\frac{2}{3}$units to the right of the pole. The directrix will be the vertical line $x=\frac{2}{3}.$