Chapter 11, Problem 34RE

To determine

To determine: The equivalent polar equation which replaces the Cartesian equation.

Answer to Problem 34RE

The required polar equation is ${\left(r\cos \theta +2\right)}^2+{\left(r\sin \theta -5\right)}^2=16$ .

Explanation of Solution

Given information:

The Cartesian equation : ${\left(x+2\right)}^2+{\left(y-5\right)}^2=16$

Calculation:

The given Cartesian equation is ${\left(x+2\right)}^2+{\left(y-5\right)}^2=16$ .

To replace the Cartesian equation by a polar equation, use the polar-rectangular equations and formulas $x=r\cos \theta$ , $y=r\sin \theta$ , $r^2=x^2+y^2$ , $\sec \theta =\frac{1}{\cos \theta }$ , $\tan \theta =\frac{\sin \theta }{\cos \theta }$ and $\tan \theta =\frac{y}{x}$ .

Rewrite the expression as:

  $\begin{array}{c}{\left(x+2\right)}^2+{\left(y-5\right)}^2=16\\ {\left(r\cos \theta +2\right)}^2+{\left(r\sin \theta -5\right)}^2=16\end{array}$

Hence, the required polar equation is ${\left(r\cos \theta +2\right)}^2+{\left(r\sin \theta -5\right)}^2=16$ .