Chapter 9.4, Problem 34E

To determine

To write the polar equation which should be programmed.

Answer to Problem 34E

The polar equation which should be programmedis $4=r\left(\theta -90°\right)$

Explanation of Solution

Given information:

The line passes through points with polar coordinates $\left(5,4\right)\text{ and }\left(15,4\right)$

Calculation:

Form the equation of line passing through $\left(5,4\right)\text{ and }\left(15,4\right)$ .

  $\begin{array}{l}\left(y-y_1\right)=\frac{y_2-y_1}{x_2-x_1}\left(x-x_1\right)\\ \left(y-4\right)=\frac{4-4}{15-5}\left(x-5\right)\\ y-4=0\left(x-5\right)\\ y-4=0\end{array}$

To convert to normal form, you need to find the value of $\pm \sqrt{A^2+B^2}$

  $\begin{array}{l}\pm \sqrt{A^2+B^2}\\ =\pm \sqrt{1+0}\\ =\pm 1\end{array}$

Since C is negative take $+1$

Rewrite the equation as

  $\frac{y}{1}-\frac{4}{1}=0$

Compare the equation to normal form that is $x\cos \phi +y\sin \phi -p=0$

Therefore,

  $p=4,\cos \phi =0,\sin \phi =1$

Now

  $\begin{array}{l}\tan \phi =\frac{\sin \phi }{\cos \phi }\\ =\frac{1}{0}\\ \phi =\arctan \left(\frac{1}{0}\right)\\ \phi =90°\end{array}$

Now substitute the value of p and $\phi$ into polar form

  $\begin{array}{l}p=r\cos \left(\theta -\phi \right)\\ 4=r\left(\theta -90°\right)\end{array}$

The polar equation is $4=r\left(\theta -90°\right)$