Chapter 6.4, Problem 46E

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 $r=\frac{2}{3\cos \theta +4\sin \theta }$ and the graph of rectangular equation is shown in figure (1).

Given data:

The equation is $3x+4y=2$ .

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 $x$ and $r\sin \theta$ for $y$ in the given equation.

  $\begin{array}{c}3r\cos \theta +4r\sin \theta =2\\ r\left(3\cos \theta +4\sin \theta \right)=2\end{array}$

Divide both sides by $3\cos \theta +4\sin \theta$ .

  $r=\frac{2}{3\cos \theta +4\sin \theta }$

The graph of the rectangular equation is a slanted line with intercepts $\left(\frac{2}{3},0\right)$ and $\left(0,\frac{1}{2}\right)$ as shown below.

  

Figure (1)

Draw the graph of the polar equation $r=\frac{2}{3\cos \theta +4\sin \theta }$ as follows.

  

Figure (2)

Therefore, the polar form of equation is $r=\frac{2}{3\cos \theta +4\sin \theta }$ and the graph of rectangular equation is shown in figure (1).