Precalculus (6th Edition) Blitzer

The required polar equation is $r=10$.
To find the polar equation substitute $x=r\cos \theta \ \text{ and }\ y=r\sin \theta$ in the provided equation, where r is the distance of that point from the origin and $\theta$ is the respective angle. So, \begin{align} & {{x}^{2}}+{{y}^{2}}=100 \\ & {{\left( r\cos \theta \right)}^{2}}+{{\left( r\sin \theta \right)}^{2}}=100 \\ & {{r}^{2}}\left( {{\cos }^{2}}\theta +{{\sin }^{2}}\theta \right)=100 \end{align} Since, ${{\cos }^{2}}\theta +{{\sin }^{2}}\theta =1$ so, \begin{align} & {{r}^{2}}\left( {{\cos }^{2}}\theta +{{\sin }^{2}}\theta \right)=100 \\ & {{r}^{2}}=100 \\ & r=10 \end{align} Therefore, the polar equation of the equation ${{x}^{2}}+{{y}^{2}}=100$ is $r=10$.