Q. 44 - Calculus

Question

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

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

Step-by-Step Solution

Answer

The graph is 

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 $x=\frac{2}{3}.$