Chapter 9.2, Problem 28AYU

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

$r\sec \theta =-4$

To determine

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

Answer to Problem 28AYU

Solution:

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

Given:

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

Explanation of Solution

Convert the polar equation to a rectangular equation $\sec \theta = - 4$ .

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 $\sec \theta = - 4$ ,

$\begin{array}{c}r\left(\frac{1}{\text{cos}\theta }\right) = - 4\\ r = - 4 \text{cos}\theta \end{array}$

Multiply both the sides by $r$,

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

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

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

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

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

To sketch:

$r \sec \theta = - 4$