Question

Find the points of intersection of the curves \(r = 2\) and \(r = 4\cos \theta .\)

Short answer

The points of intersection of the curves \(r = 2\) and \(r = 4\cos \theta \) is \(\left( {2, \pm \frac{\pi }{3}} \right).\)

Step-by-step solution

Definition of the parametric equation

A parametric equation in mathematics specifies a set of numbers as functions of one or more independent variables known as parameters.

Calculating the points of intersection

It is given that,\(r = 2,r = 4\cos \theta \).

Both polar equations represent circles and we can find the point(s) of intersection by method of substitution.

From the first equation, we know that the solution for\(r\)is given by\(r = 2.\)

Substitute this to the second equation\(r = 4\cos \theta \)to obtain the following equation:

\(\begin{aligned}{c}2 = 4\cos \theta \\2 - 4\cos \theta = 0\\1 - 2\cos \theta = 0\\\cos \theta = - \frac{1}{2}\end{aligned}\)

The period of\(\cos \theta \)is\(2\pi \)and over the interval\( - \pi \le \theta \le \pi \), the solutions of this equation are:

\(\theta = - \frac{\pi }{3},\frac{\pi }{3}\)

Which, are the quadrant IV and quadrant I angles, respectively, where\(\cos \theta > 0\).

Therefore, the points of intersection (in polar coordinates) of the curves are

\(\left( {2, \pm \frac{\pi }{3}} \right).\)