Chapter 9.5, Problem 53E

To determine

To calculate: To convert the rectangular equation to polar form

Answer to Problem 53E

The polar equation is $r=2a\cos \theta$

Explanation of Solution

Given information: Rectangular equation is $x^2+y^2-2ax=0$

Formula Used:

Polar coordinate of a point $\left(r,\theta \right)$ in rectangular form is given as

  $\begin{array}{l}x=r\cos \theta \\ y=r\sin \theta \end{array}$

Calculation:

Rectangular equation is given as

  $x^2+y^2-2ax=0$

Substituting $x=r\cos \theta ;y=r\sin \theta$ in above equation:

  $\begin{array}{l}{\left(r\cos \theta \right)}^2+{\left(r\sin \theta \right)}^2-2a\left(r\cos \theta \right)=0\\ r^2{\cos }^2\theta +r^2{\sin }^2\theta -2ar\cos \theta =0\\ r^2\left({\cos }^2\theta +{\sin }^2\theta \right)-2ar\cos \theta =0\\ r^2\left(1\right)-2ar\cos \theta =0\\ r^2-2ar\cos \theta =0\\ r^2=2ar\cos \theta \\ r=2a\cos \theta \end{array}$

Thus, polar equation is $r=2a\cos \theta$

Conclusion:

Hence, polar equation is $r=2a\cos \theta$