3.6 Normal form

1.1 Angle of inclination of a line
1.2 Slope or Gradient of a line

3.1 Point-slope form
3.2 Slope-Intercept form
3.3 Two point form
3.4 Determinant form
3.5 Intercept form
3.6 Normal form
3.7 Parametric form

7.1 Reflection with respect to $X-$axis
7.2 Reflection with respect to $Y-$axis
7.3 Reflection with respect to Origin
7.4 Reflection with respect to the line $x=a$
7.5 Reflection with respect to the line $y=b$
7.6 Reflection with respect to the line $y=x$

8.1 Acute (internal) angle bisector and Obtuse (external) angle bisector

The equation of straight line where the length of perpendicular from the origin to the line is $p$ and this perpendicular makes an angle $\alpha $ with positive $X-$axis is written as$$x\cos \alpha + y\sin \alpha = p$$ where $p > 0,{\text{ }}0 \leqslant \alpha < 2\pi $.

Proof: Let $AB$ be a line such that the length of perpendicular from $O$ to the line be $p$ i.e., $ON=p$ and $\angle NOQ = \alpha $.

Let $P(x,y)$ be any point on the line. Draw $PL$ perpendicular from $P$ on $x-$axis. Let line $AB$ cuts $x$ and $y$ axes at $Q$ and $R$ respectively.

In $\Delta PLQ$, $$\begin{equation} \begin{aligned} \tan \alpha = \frac{{LQ}}{{PL}} = \frac{{LQ}}{y} \\ LQ = y\tan \alpha \\\end{aligned} \end{equation} $$

Also, In $\Delta ONQ$, $$\begin{equation} \begin{aligned} \cos \alpha = \frac{{ON}}{{OQ}} = \frac{p}{{OL + LQ}} \\ OL\cos \alpha + LQ\cos \alpha = p \\ x\cos \alpha + y\tan \alpha \cos \alpha = p{\text{ }}(\because OL = x{\text{ and }}LQ = y\tan \alpha ) \\ x\cos \alpha + y\sin \alpha = p \\\end{aligned} \end{equation} $$

which is the required equation of line $AB$.