# Normal Form of a Line

If $p$ is the length of perpendicular from origin to the non-vertical line $l$ and $\alpha$ is the inclination of $p$, then show that the equation of the line is

To prove this equation of straight in normal form, Let $P\left( {x,y} \right)$ be any point on the straight line $l$. Since the line intersects the coordinate axes at points $A$ and $B$, so $OA$ and $OB$ becomes its X-intercept and Y-intercept as shown in the given diagram. Now using equation of straight line intercepts form, we have

If $C$ is the foot of the perpendicular draw from origin $O$ to the non-vertical straight line, then consider $OCA$ is the right triangle as given in the diagram, so using the trigonometric ratio $\cos \alpha$ as follows

Since $OCB$ is a right triangle, so $\frac{{OC}}{{OB}} = \cos \left( {{{90}^ \circ } - \alpha } \right)$

Now the putting the values of $OA$ and $OB$ in equation (i), we get

Which is the equation of straight line in normal form.