Q 14 - Calculus

Question

Sketch the graphs of the equations

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

What is the relationship between these graphs? What is the eccentricity of each graph?

Step-by-Step Solution

Answer

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 $2$.

Step 1. Given Information

We have given the following two equations :-

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

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 :-

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

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 $r=\frac{2}{1+2\cos \theta }$ is open left and right. Also the graph of hyperbola $r=\frac{2}{1+2\sin \theta }$ 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 $\frac{1}{r}=1+e\cos \theta$, where $e$ is the eccentricity.

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