Chapter 6, Problem 68RE

Solution

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

The polar form of given rectangular equation is $r=\frac{4}{2\cos \theta -3\sin \theta }$ and the graph of the polar equation is a straight line.

Given data:

The rectangular equation is $2x-3y=4$ .

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:

  $2x-3y=4$

Substitute $r\cos \theta$ for $x$ and $r\sin \theta$ for $y$ .

  $\begin{array}{c}2r\cos \theta -3r\sin \theta =4\\ r=\frac{4}{2\cos \theta -3\sin \theta }\end{array}$

Therefore, the polar form of given equation is $r=\frac{4}{2\cos \theta -3\sin \theta }$ .

Plot the graph of equation.

  

Figure (1)

From the graph, it can be observed that the graph of polar equation $r=\frac{4}{2\cos \theta -3\sin \theta }$ is a straight line.

Therefore, polar form of given rectangular equation is $r=\frac{4}{2\cos \theta -3\sin \theta }$ and the graph of the polar equation is a straight line.