Question

How do you write a polar equation for the circle centered at the origin with radius $\sqrt{2}$?

Answer 1

Write a polar equation for the circle:

The standard form for the equation of a circle centered at the origin is $x^2+y^2=a^2$.

Here, $a$ is radius of the circle.

Given radius of the circle is $a=\sqrt{2}$.

The equation of a circle becomes $x^2+y^2=2$

Substitute polar coordinates of $\left(x,y\right)=\left(r\cos \theta ,r\sin \theta \right)$ in $x^2+y^2=2$.

$\begin{array}{rcl}{x}^2+y^2& =& 2\\ & \Rightarrow & {\left(r\cos \theta \right)}^2+{\left(r\sin \theta \right)}^2=2\\ & \Rightarrow & r^2\left({\cos }^2\theta +{\sin }^2\theta \right)=2\\ & \Rightarrow & r^2=2\end{array}$

Hence, a polar equation for the circle centered at the origin with radius $\sqrt{2}$ is $r^2=2$.