Q 67 - Calculus

Question

Show that the graph of the equation $r=k\cos \theta$ is a circle tangent to the y-axis for any $k\ne 0$. What are the center and radius of the circle? 

Step-by-Step Solution

Answer

The graph of the equation $y=k\cos \theta$ is a circle whose tangent is at y=axis, where $\frac{k}{2}$ is the radius of circle and $\left(\frac{k}{2},0\right)$ is the center of the circle

Step 1. Given Information

We have the following equation in polar coordinates :-

$y=k\cos \theta$.

We have to prove that this is the equation of circle. Also we have to find center and radius of the circle.

Step 2. Required proof

The given equation is :-

$y=k\cos \theta$.

If we take the value of the $k$ as $k=2$ and graph the equation, then we have the following graph :-

Here we can see that the graph of the equation is the circle with tangent at y-axis. Here the center of the radius is $1$ and center is $\left(1,0\right)$.

We will get similar results for other values of $k$ as well.

Then we can conclude that the graph of the given equation $y=k\cos \theta$ is a circle with center at origin. Also the center of the circle is $\left(\frac{k}{2},0\right)$ and radius is $\frac{k}{2}$.