JEE Main & Advanced Mathematics Straight Line General Equation of a Straight Line and its Transformation in Standard Forms

General form of equation of a line is \[ax+by+c=0\], its

(1) Slope intercept form: \[y=-\frac{a}{b}x-\frac{c}{b}\], slope \[m=\frac{a}{b}\] and intercept on y-axis is, \[C=-\frac{c}{b}\].

(2) Intercept form : \[\frac{x}{-c/a}+\frac{y}{-c/b}=1\], \[x\] intercept is \[=\left( -\frac{c}{a} \right)\] and y intercept is \[=\left( -\frac{c}{b} \right)\].

(3) Normal form : To change the general form of a line into normal form, first take c to right hand side and make it positive, then divide the whole equation by \[\sqrt{{{a}^{2}}+{{b}^{2}}}\] like

\[-\frac{ax}{\sqrt{{{a}^{2}}+{{b}^{2}}}}-\frac{by}{\sqrt{{{a}^{2}}+{{b}^{2}}}}=\frac{c}{\sqrt{{{a}^{2}}+{{b}^{2}}}},\]

where \[\cos \alpha =-\frac{a}{\sqrt{{{a}^{2}}+{{b}^{2}}}},\,\,\sin \alpha =-\frac{b}{\sqrt{{{a}^{2}}+{{b}^{2}}}},\,\,p=\frac{c}{\sqrt{{{a}^{2}}+{{b}^{2}}}}\]