# Chapter 6 - Section 6.3 - Polar Coordinates - Exercise Set - Page 743: 70

The rectangular equation is, ${{x}^{2}}+{{\left( y+2 \right)}^{2}}=4$

#### Work Step by Step

Using, ${{r}^{2}}={{x}^{2}}+{{y}^{2}}$, we will convert the polar equation to the rectangular equation. Therefore, \begin{align} & r=-4\sin \theta \\ & {{r}^{2}}=-4r\sin \theta \\ & {{x}^{2}}+{{y}^{2}}=-4r\sin \theta \\ & =-4y \end{align} We can further simplify it by completing the square on y as, \begin{align} & {{x}^{2}}+{{y}^{2}}=-4y \\ & {{x}^{2}}+{{y}^{2}}+4y+4=4 \\ & {{x}^{2}}+{{\left( y+2 \right)}^{2}}=4 \end{align}

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