Q 15 - Calculus

Question

Sketch the graphs of the equations

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

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 ellipse with focus at origin. One has major axis x-axis and other has y-axis.

The eccentricity of both graphs is $\frac{1}{2}$.

Step 1. Given Information

We have given the following two equations :-

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

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 :-

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

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 ellipses. The graph of ellipse $r=\frac{2}{2+\cos \theta }$ has x-axis as the major axis and the graph of $r=\frac{2}{2+\sin \theta }$ has y-axis as the major x-axis. The focus of both ellipses at origin.

Step 4. Eccentricity of graphs

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

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

The eccentricity of both ellipses is $\frac{1}{2}$.