Chapter 9, Problem 18RE

To determine

To write:the polar equation of the rectangular coordinates $\left(x,y\right)$ of the given variables $r\text{ and }\theta$ . Also, identify the equation and graph it.

Answer to Problem 18RE

The equation of the circle is ${\left(x-4\right)}^2+{\left(y+2\right)}^2=25$ .

Explanation of Solution

Given:

The givenequation is $r^2+4r\sin \theta -8r\cos \theta =5$ .

Calculation:

As it is known that $x=r\cos \theta y=r\sin \theta$ .

The given equation is expresses as:

  $\begin{array}{l}{r}^2+4r\sin \theta -8r\cos \theta =5\\ \therefore x^2+y^2+4y-8x=5\\ \Rightarrow x^2-8x+16+y^2+4y+4=5+16+4(\text{add }16+4\text{on both the sides})\\ \therefore {\left(x-4\right)}^2+{\left(y+2\right)}^2=25(usingidentity:{\left(a\pm b\right)}^2=a^2\pm 2ab+b^2)\end{array}$

From above, it is found that the above equation is the equation of the circle.

Now, draw the graph of the circle.

  

Hence, polar equation of the rectangular coordinates is ${\left(x-4\right)}^2+{\left(y+2\right)}^2=25$ .