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Q 14

Expert-verified
Found in: Page 772

### Calculus

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

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# Sketch the graphs of the equations$r=\frac{2}{1+2\mathrm{cos}\theta }$ and $r=\frac{2}{1+2\mathrm{sin}\theta }$What is the relationship between these graphs? What is the eccentricity of each graph?

The graphs of the given equations are as following :-

Both the graphs are hyperbola with one focus at origin. Open opens to left and right and other opens to up and down.

Also the eccentricity of both hyperbolas is $2$.

See the step by step solution

## Step 1. Given Information

We have given the following two equations :-

$r=\frac{2}{1+2\mathrm{cos}\theta }$ and $r=\frac{2}{1+2\mathrm{sin}\theta }$

We have to draw the graph of these equations. We have to find the relationship between the graphs. Also we have to find the eccentricity of each graph.

## Step 2. Draw graphs of the equations

The given two equations are :-

$r=\frac{2}{1+2\mathrm{cos}\theta }$ and $r=\frac{2}{1+2\mathrm{sin}\theta }$

We can draw the graph of these equations as following :-

## Step 3. Find relationship between the graphs :-

We draw the graphs of given equations as following :-

We can see that both the graphs are hyperbolas. That graph of hyperbola $r=\frac{2}{1+2\mathrm{cos}\theta }$ is open left and right. Also the graph of hyperbola $r=\frac{2}{1+2\mathrm{sin}\theta }$ is open up and down.

One focus of both hyperbolas is origin.

## Step 4. Eccentricity of graphs

From the graph we can see that the both equations are parabolas.

Compare the given equations with the equation $\frac{1}{r}=1+e\mathrm{cos}\theta$, where $e$ is the eccentricity.

Then we can find that the eccentricity of graphs is $2$.

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