Chapter 10.9, Problem 10E

To determine

To match: The polar equation $r=\frac{3}{2+\cos \theta }$ with its graph.

Answer to Problem 10E

The polar graph represented by $r=\frac{3}{2+\cos \theta }$ is an ellipse and is matched with $\left(d\right)$ .

  

Explanation of Solution

Given information:

The polar equation $r=\frac{3}{2+\cos \theta }$ .

Consider the polar equation $r=\frac{3}{2+\cos \theta }$ .

When polar equations are either of the form $r=\frac{ep}{1\pm e\sin \theta }$ or $r=\frac{ep}{1\pm e\cos \theta }$ then it is called a conic. Here $e>0$ is the eccentricity and $\left|p\right|$ is the distance between the focus also called pole and the directrix.

When $0<e<1$ , conic is called ellipse.

When $e=1$ , conic is called parabola.

When $e>1$ , conic is called hyperbola.

For $r=\frac{3}{2+\cos \theta }$ . Divide numerator and denominator by 2 and rewrite the equation as,

  $\begin{array}{c}r=\frac{3}{2+\cos \theta }\\ =\frac{\frac{3}{2}}{1+\frac{1}{2}\cos \theta }\\ =\frac{3\left(\frac{1}{2}\right)}{1+\frac{1}{2}\cos \theta }\end{array}$

Therefore, eccentricity is $e=\frac{1}{2}$ . Since, $0<e<1$ , conic is called ellipse.

Thu, the polar graph represented by $r=\frac{3}{2+\cos \theta }$ is an ellipse and is matched with $\left(d\right)$ .