Chapter 8.5, Problem 16E

Solution

To find:The graph for the given polar equation and the viewing window of the graph.

Thethe conic is aellipse, and graph (d) with the viewing window of $\left[-5,5\right]$ by $\left[-3,3\right]$ is represented by the polar equation $r=\frac{4}{3+2\cos \theta }$ .

Given information:

The given polar equation is $r=\frac{4}{3+2\cos \theta }$ .

Formula used:

  $e<1$ , the conic is an ellipse.

  $e=1$ , the conic is a parabola.

  $e>1$ , the conic is a hyperbola.

  $r=\frac{ke}{1+e\cos \theta }$ when directrix is vertical, to the right of the pole: $x=k$ .

  $r=\frac{ke}{1-e\cos \theta }$ when directrix is vertical, to the right of the pole: $x=-k$ .

  $r=\frac{ke}{1+e\sin \theta }$ when directrix is horizontal, to the right of the pole: $y=k$ .

  $r=\frac{ke}{1-e\sin \theta }$ when directrix is horizontal, to the right of the pole: $y=-k$ .

Calculation:

Compare $r=\frac{4}{3+2\cos \theta }$ with $r=\frac{ke}{1+e\cos \theta }$ .

Take $3$ common in the denominator.

  $r=\frac{\frac{4}{3}}{1+\frac{2}{3}\cos \theta }$

Here, $k=2$ and $e=\frac{2}{3}$ .

Since, $e<1$ , the conic is aellipse.

The equation of directrix is $x=k$ .

Substitute $2$ for $k$ in $x=k$ .

Thus, equation of directrix for $r=\frac{4}{3+2\cos \theta }$ is $x=2$ .

This matches the graph (d).

  

The viewing window of the graph is $\left[-5,5\right]$ by $\left[-3,3\right]$ .

Therefore, the conic is a hyperbola, and graph (d) with the viewing window of $\left[-5,5\right]$ by $\left[-3,3\right]$ is represented by the polar equation $r=\frac{4}{3+2\cos \theta }$ .