Converting Linear Equations in Standard Form to Normal Form

The general equation or standard equation of a straight line is:

ax + by + c = 0

Where a and b are constants and either a \ne 0 or b \ne 0.

Convert the standard equation of line ax + by + c = 0 into the normal form

x\cos \alpha + y\sin \alpha = p

The procedure of converting the equation into the normal form is as follows:

\begin{gathered} ax + by + c = 0 \\ \Rightarrow - ax - by = c \\ \end{gathered}

 

Dividing both sides of the equation by  \pm \sqrt {{a^2} + {b^2}}  we have

 \Rightarrow - \frac{a}{{ \pm \sqrt {{a^2} + {b^2}} }}x - \frac{b}{{ \pm \sqrt {{a^2} + {b^2}} }}y = \frac{c}{{ \pm \sqrt {{a^2} + {b^2}} }}

 

This is the equation of the line in normal form, where \frac{c}{{ \pm \sqrt {{a^2} + {b^2}} }} is the length of the normal form origin of the line. Since the length of the normal form origin to the line must be positive, so:

(i)  - \frac{a}{{\sqrt {{a^2} + {b^2}} }}x - \frac{b}{{\sqrt {{a^2} + {b^2}} }}y = \frac{c}{{\sqrt {{a^2} + {b^2}} }} is the normal form of the line if c > 0.

(ii) \frac{a}{{\sqrt {{a^2} + {b^2}} }}x + \frac{b}{{\sqrt {{a^2} + {b^2}} }}y = \frac{c}{{ - \sqrt {{a^2} + {b^2}} }} is the normal form of line if c < 0.

 

Example: Convert the equation 2x + 5y - 6 = 0 into normal form.

The equation of a line in standard form is 2x + 5y - 6 = 0

 \Rightarrow 2x + 5y = 6

Dividing both sides of the equation by the normalizing factor \sqrt {{{\left( 2 \right)}^2} + {{\left( 5 \right)}^2}} = \sqrt {29} , we have

\Rightarrow \frac{{2x}}{{\sqrt {29} }} + \frac{{5y}}{{\sqrt {29} }} = \frac{6}{{\sqrt {29} }}

Compare with the normal form of a line x\cos \alpha + y\sin \alpha = p, where \cos \alpha = \frac{2}{{\sqrt {29} }}, \sin \alpha = \frac{5}{{\sqrt {29} }}and p = \frac{6}{{\sqrt {29} }}.