Chapter 9.4, Problem 3CFU

To determine

To find: the angle and length of the normal for a rectangular equation and the polar form of the equation.

Answer to Problem 3CFU

  $r=k\sec \theta$ .

Explanation of Solution

Given:

The rectangular equation of the line is:

  $x=k$ .

Concept used:

The polar form of a complex number is another way to represent a complex number. The form

$z=x+iy$ is rectangular coordinate and in polar from $x=r\cos \theta ,y=r\sin \theta$ and the polar form of equation:

  $z=r\left(\cos \theta +i\sin \theta \right)$ .

Calculation:

The rectangular equation of the line is:

  $x=k\mathrm{......}\left(i\right)$

It has the polar form of equation of a line is:

  $x\cos \varphi +y\sin \varphi -p=\mathrm{0..........}\left(ii\right)$ .

Where, $p$ is the perpendicular distance of the line from the origin and $\varphi$ is the angle made by the perpendicular with the positive $x$ axis.

It rewrites the equation $\left(i\right)$ in normal form as:

$\begin{array}{l}x=k.\\ \Rightarrow x\left(\cos 0^0\right)+y\left(\sin 0^0\right)-k=\mathrm{0.........}\left(iii\right)\end{array}$

Comparing $\left(ii\right)\text{ and }\left(iii\right)$ :

  $\varphi =0^0$

To convert the equation $\left(i\right)$ to polar equation it can be replace by $x=r\cos \theta$ in

  $\begin{array}{l}r\cos \theta =k.\\ r=k\sec \theta .\end{array}$

Hence, the polar form of equation $\left(i\right)$ is:

  $r=k\sec \theta$ .