Converting Linear Equations in Standard Form to Normal Form

The general equation or standard equation of a straight line is given by

ax + by + c = 0


Where a and b are any constants and also 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 into normal form as follows

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


Dividing on both sides  \pm \sqrt {{a^2} + {b^2}} of the equation, 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 equation of line in normal form, where \frac{c}{{ \pm \sqrt {{a^2} + {b^2}} }} is the length of normal form the 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 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. We have equation of line in standard form is 2x + 5y - 6 = 0

 \Rightarrow 2x + 5y = 6

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

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

Compare with 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} }}.

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