Chapter 6, Problem 64RE

Solution

To find: The rectangular form of the polar equation and identify the graph.

The rectangular form of given equation is $x=3$ and the graph of the equation is a straight line parallel to y-axis.

Given data:

The polar equation is $r=3\sec \theta$ .

Formula used:

Let $\left(r,\theta \right)$ and $\left(x,y\right)$ are the polar and rectangular coordinates of the point P.

Then,

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

Calculation:

  $r=3\sec \theta$

Simplify above equation.

  $\begin{array}{c}r=3\times \frac{1}{\cos \theta }\\ r\cos \theta =3\end{array}$

Substitute $x$ for $r\cos \theta$ .

  $x=3$

Therefore, the rectangular equation is $x=3$ .

Plot the graph of equation.

  

Figure (1)

From the graph, it can be observed that the graph of the equation $x=3$ is a straight line parallel to y-axis.

Therefore, the rectangular form of given equation is $x=3$ and the graph of the equation is a straight line parallel to y-axis.