Chapter 9.5, Problem 88E

To determine

To convert the polar equation $r=\cos \theta +3\sin \theta$ into rectangular form and hence find the centre and radius of the circle.

Answer to Problem 88E

The centre and the radius of the circle is $\left(\frac{1}{2},1\right)$ and $\frac{\sqrt{5}}{2}$ .

Explanation of Solution

Given information: $r=\cos \theta +3\sin \theta$ .

Concept used:

1. To convert rectangular coordinates into polar coordinates,

replace $(x,y)$ with $(r\cos \theta ,r\sin \theta )$ such that $r=\sqrt{x^2+y^2}$ and $\theta ={\tan }^{-1}\left(\frac{y}{x}\right)$ .

2. The general equation of circle in rectangular form is $x^2+y^2+2ax+2by+c=0$ .

The centre and the radius of the circle is $(-a,-b)$ and $\sqrt{a^2+b^2-c}$ .

Given,  $r=\cos \theta +3\sin \theta$ .

  $\begin{array}{l}{r}^2=r(\cos \theta +3\sin \theta )\\ r^2=x+2y\\ x^2+y^2-x-2y=0\\ x^2+y^2-2\left(\frac{1}{2}\right)x-2y=0,\text{which}\text{is}\text{the}\text{general}\text{equation}\text{of}\text{the}\text{circle}\end{array}$

The centre and the radius of the circle is $\left(\frac{1}{2},1\right)$ and $\sqrt{\frac{1}{4}+1}=\frac{\sqrt{5}}{2}$ .