Q. 45 - 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 will be

Step 1. Given information.

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

Step 2. Comparison.

On comparing the given equation with 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.

For $r=\frac{2}{1-3\cos \theta }$ the eccentricity is$e=-3,$ so the graph is a hyperbola. The directrix will be the vertical line $x=-\frac{2}{3}$