Chapter 6.4, Problem 44E

Solution

To convert: The rectangular equation to polar form, sketches the graph of the rectangular equation, and supports the answer by graphing the polar equation with grapher.

The polar form of equation is $r=5\sec \theta$ and the graph of rectangular equation is shown in figure (1).

Given data:

The equation is $x=5$ .

Formula used:

Use the following conversion relationships between a polar coordinate $\left(r,\theta \right)$ and its corresponding rectangular coordinate $\left(x,y\right)$ .

  $\begin{array}{c}x=r\cos \theta \\ y=r\sin \theta \\ r^2=x^2+y^2\\ \tan \theta =\frac{y}{x}\end{array}$

Calculation:

Substitute $r\cos \theta$ for $x$ in the given equation.

  $r\cos \theta =5$

Divide both sides by $\cos \theta$ .

  $\begin{array}{c}r=\frac{5}{\cos \theta }\\ r=5\sec \theta \end{array}$

The graph of the rectangular equation is a vertical line passing through $\left(5,0\right)$ as shown below.

  

Figure (1)

Draw the graph of the polar equation $r=5\sec \theta$ as follows.

  

Figure (2)

Therefore, the polar form of equation is $r=5\sec \theta$ and the graph of rectangular equation is shown in figure (1).