JEE Main & Advanced Mathematics Straight Line Equations of Straight Line in Different Forms

(1) Slope form : Equation of a line through the origin and having slope \[m\] is \[y=mx\].

(2) One point form or Point slope form : Equation of a line through the point \[({{x}_{1}},{{y}_{1}})\,\]and having slope \[m\] is \[y-{{y}_{1}}=m\,(x-{{x}_{1}})\].

(3) Slope intercept form  : Equation of a line (non-vertical) with slope m and cutting off an intercept \[c\] on the y-axis is \[y=mx+c\].

The equation of a line with slope m and the x-intercept \[d\] is \[y=m(x-d)\].

(4) Intercept form : If a straight line cuts x-axis at A and the y-axis at B then OA and OB are known as the intercepts of the line on x-axis and y-axis respectively.

Then, equation of a straight line cutting off intercepts \[a\] and \[b\] on x–axis and y–axis respectively is \[\frac{x}{a}+\frac{y}{b}=1\].

If given line is parallel to \[X\] axis, then X-intercept is undefined.

If given line is parallel to \[Y\] axis, then Y-intercept is undefined.

(5) Two point form: Equation of the line through the points  \[A({{x}_{1}},{{y}_{1}})\,\]  and \[B({{x}_{2}},{{y}_{2}})\] is, \[(y-{{y}_{1}})=\frac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}(x-{{x}_{1}})\].

In the determinant form it is gives as \[\left| \,\begin{matrix} x & y & 1  \\ {{x}_{1}} & {{y}_{1}} & 1  \\  {{x}_{2}} & {{y}_{2}} & 1  \\ \end{matrix}\, \right|=0\]  is the equation of line.

(6) Normal or perpendicular form : The equation of the straight line upon which the length of the perpendicular from the origin is \[p\] and this perpendicular makes an angle \[\alpha \] with x-axis is \[x\cos \alpha +y\sin \alpha =p\].

(7) Symmetrical or parametric or distance form of the line :  Equation of a line passing through \[({{x}_{1}},{{y}_{1}})\] and making an angle q with the positive direction of x-axis is \[\frac{x-{{x}_{1}}}{\cos \theta }=\frac{y-{{y}_{1}}}{\sin \theta }=\pm r\], where r is the distance between the point \[P\,(x,\,\,y)\]  and \[A({{x}_{1}},{{y}_{1}})\].

The co-ordinates of any point on this line may be taken as \[({{x}_{1}}\pm r\cos \theta ,{{y}_{1}}\pm r\sin \theta )\], known as parametric co-ordinates. \['r'\]  is called the parameter.