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Q. 51

Expert-verified
Found in: Page 772

### Calculus

Book edition 1st
Author(s) Peter Kohn, Laura Taalman
Pages 1155 pages
ISBN 9781429241861

# Use polar coordinates to graph the conics in Exercises 44–51. $r=\frac{2}{2+\mathrm{sin}\theta }$

The graph is

See the step by step solution

## Step 1. Given information.

The given equation is $r=\frac{2}{2+\mathrm{sin}\theta }$.

## Step 2. Comparison.

On comparing the given equation with standard form $r=\frac{eu}{1+e\mathrm{sin}\theta }$.

$r=\frac{\frac{2}{2}}{\frac{2+\mathrm{sin}\theta }{2}}\phantom{\rule{0ex}{0ex}}r=\frac{1}{1+\frac{1}{2}\mathrm{sin}\theta }\phantom{\rule{0ex}{0ex}}\text{Here}eu=1\phantom{\rule{0ex}{0ex}}\text{The eccentricity}e=\frac{1}{2}\phantom{\rule{0ex}{0ex}}u=2$

## Step 3. Graph.

For $r=\frac{2}{2+\mathrm{sin}\theta }$the eccentricity is $e=\frac{1}{2}$, so the graph is an ellipse. The directrix is parallel to the polar axis at a distance $u=2$ units above the pole. The directrix is $2$ units above the pole .