Q 13 - Calculus

Question

Sketch the graphs of the equations

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

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 parabolas with focus at origin. One opens to left and other is downward.

Also the eccentricity of parabolas is always $1$.

Step 1. Given Information

We have given the following two equations :-

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

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 :-

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

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 the both graphs are parabolas.

The graph of $r=\frac{1}{1+\cos \theta }$ opens to the left and the graph of $r=\frac{1}{1+\sin \theta }$ is downward parabola.

Also the focus of both parabolas is at origin.

Step 4. Eccentricity of graphs

We see that the both graphs are parabolas. Also we know that the eccentricity of each and every parabola is one.

So the eccentricity of both graphs is $1$.