Chapter 9.2, Problem 23AYU

In Problems 15-30, transform each polar equation to an equation in rectangular coordinates. Then identify and graph the equation.

$r=-4\sin \theta$

To determine

To find: Transform the polar equation $r = - 4 \text{sin}\theta$ to rectangular coordinates. Then identify and graph the equation.

Answer to Problem 23AYU

Solution:

The graph of $r = - 4 \text{sin}\theta$ is the equation of circle $x^2 + {\left(\text{y} + 2\right)}^2 = 2^2$ with center $\left(0, - 2\right)$ and radius 2.

Given:

It is asked to transform the polar equation $r = - 4 \text{sin}\theta$ to rectangular coordinates and graph it.

Explanation of Solution

Convert the polar equation to a rectangular equation.

$r = - 4 \text{sin}\theta$

Formula used:

$\begin{array}{c}\left(x, y\right) = \left(r \text{cos}\theta , r \text{sin}\theta \right)\\ x^2 + y^2 = {\text{r}}^2\end{array}$

Consider $r = - 4 \text{sin}\theta$,

Multiply both the sides by $r$,

$\begin{array}{c}{r}^2 = - 4r \text{sin}\theta \\ x^2 + y^2 = - 4y\\ x^2 +y^2 + 4y = 0\end{array}$

Now use completion of square for ‘$y$’.

$\begin{array}{c}{x}^2 + y^2 + 4y + 4 = 4\\ x^2 + {(\text{y} + 2)}^2 = 2^2\end{array}$

The above is an equation of circle with center $\left(0, - 2\right)$ and radius 2.

The graph of $r = - 4 \text{sin}\theta$ is the equation of circle $x^2 + {\left(\text{y} + 2\right)}^2 = 2^2$ with center $\left(0, - 2\right)$ and radius 2.

To sketch:

$r = - 4 \text{sin}\theta$