Converting Linear Equation in Standard Form to Symmetric 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 $b \ne 0$.

Putting $y = 0$ in the above standard equation of a line, we have

$$\begin{gathered} ax + b\left( 0 \right) + c = 0 \\ \Rightarrow ax = - c \\ \Rightarrow x = - \frac{c}{a} \\ \end{gathered}$$

This shows that the line is passing through the point $\left( { - \frac{c}{a},0} \right)$.

Since we have $\tan \alpha = m = - \frac{a}{b}$, by squaring ${\tan ^2}\alpha = {\left( {\frac{a}{b}} \right)^2} \Rightarrow {\tan ^2}\alpha = \frac{{{a^2}}}{{{b^2}}}$

$$\begin{gathered} \Rightarrow {\sec ^2}\alpha - 1 = \frac{{{a^2}}}{{{b^2}}} \\ \Rightarrow {\sec ^2}\alpha = \frac{{{a^2}}}{{{b^2}}} + 1 \\ \Rightarrow {\cos ^2}\alpha = \frac{{{b^2}}}{{{a^2} + {b^2}}} \\ \Rightarrow \cos \alpha = \pm \frac{b}{{\sqrt {{a^2} + {b^2}} }} \\ \end{gathered}$$

$$\begin{gathered} \Rightarrow {\sin ^2}\alpha = 1 - {\cos ^2}\alpha \\ \Rightarrow {\sin ^2}\alpha = 1 - \frac{{{b^2}}}{{{a^2} + {b^2}}} \\ \Rightarrow {\sin ^2}\alpha = \frac{{{a^2} + {b^2} - {b^2}}}{{{a^2} + {b^2}}} \\ \Rightarrow {\sin ^2}\alpha = \frac{{{a^2}}}{{{a^2} + {b^2}}} \\ \Rightarrow \sin \alpha = \pm \frac{a}{{\sqrt {{a^2} + {b^2}} }} \\ \end{gathered}$$

Since the equation of a straight line in symmetric form is

$$\frac{{x - {x_1}}}{{\cos \alpha }} = \frac{{y - {y_1}}}{{\sin \alpha }}$$

Now we put ${x_1} = - \frac{c}{a},\,\,{y_1} = 0,\,\,\cos \alpha = \pm \frac{b}{{\sqrt {{a^2} + {b^2}} }},\,\,\sin \alpha = \pm \frac{a}{{\sqrt {{a^2} + {b^2}} }}$ in the standard equation of symmetric form, and we have

$$\begin{gathered} \frac{{x - \left( { - \frac{c}{a}} \right)}}{{ \pm \frac{b}{{\sqrt {{a^2} + {b^2}} }}}} = \frac{{y - 0}}{{ \pm \frac{a}{{\sqrt {{a^2} + {b^2}} }}}} \\ \Rightarrow \frac{{x + \frac{c}{a}}}{{ \pm \frac{b}{{\sqrt {{a^2} + {b^2}} }}}} = \frac{y}{{ \pm \frac{a}{{\sqrt {{a^2} + {b^2}} }}}} \\ \end{gathered}$$

This is the equation of a line in symmetric form transformed from its general form or standard form of a straight line.