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Q. 44

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

The graph is

See the step by step solution

## Step 1. Given information.

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

## Step 2. Comparison.

On comparing the given equation with the standard form,

$r=\frac{eu}{1+e\mathrm{cos}\theta }\phantom{\rule{0ex}{0ex}}\text{The eccentricity}e=3\phantom{\rule{0ex}{0ex}}\text{Here}eu=2\phantom{\rule{0ex}{0ex}}3·u=2⇒u=\frac{2}{3}$

## Step 3. Graph of the equation.

For $r=\frac{2}{1+3\mathrm{cos}\theta }$ 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 $x=\frac{2}{3}.$