Chapter 8.2, Problem 62E

To determine

To calculate: To compare the polar equation of the circle $r=2$ with its rectangular equation of the line $y=2$ with its polar equation and find which coordinate system is simpler and to find which coordinate system one would choose to study lines.

Answer to Problem 62E

Polar coordinate system is simpler and easier to study circles

Explanation of Solution

Given: Polar equation of circle is $r=2$

Formula Used:

A polar equation is any equation that describes a relation between r and θ , where r represents the distance from the pole (origin) to a point on a curve, and θ represents the counter-clockwise angle made by a point on a curve, the pole, and the positive x-axis.

Calculation:

Given polar equation of circle as follows:

  $r=2$

Rectangular equations for the above polar equation is given as

  $x^2+y^2=r^2$

Substituting the values,

  $\begin{array}{l}{x}^2+y^2=2^2\\ x^2+y^2=4\end{array}$

Now, in this case, polar equation is simpler as one can write down the equation with just one variable, while the rectangular form of equation comprises of two variables

Although, rectangular equation can be solved for one variable, but the resulting equation is slightly complicated

  $\begin{array}{l}{y}^2=4-x^2\\ y=\pm \sqrt{4-x^2}\end{array}$

Hence, polar coordinate system is much simpler to solve as well as study the circles.

Conclusion:

Hence, Polar coordinate system is simpler and easier to study lines