Slope \((m)\) of a non-vertical line passing through the points \(\left(x_1, y_1\right)\) and \(\left(x_2, y_2\right)\) is given by \(m=\frac{y_2-y_1}{x_2-x_1}=\frac{y_1-y_2}{x_1-x_2}, \quad x_1 \neq x_2\).
If a line makes an angle รก with the positive direction of \(x\)-axis, then the slope of the line is given by \(m=\tan \alpha, \alpha \neq 90^{\circ}\).
Slope of horizontal line is zero and slope of vertical line is undefined.
An acute angle (say \(\theta\) ) between lines \(\mathrm{L}_1\) and \(\mathrm{L}_2\) with slopes \(m_1\) and \(m_2\) is given by \(\tan \theta=\left|\frac{m_2-m_1}{1+m_1 m_2}\right|, 1+m_1 m_2 \neq 0\).
Two lines are parallel if and only if their slopes are equal.
Two lines are perpendicular if and only if product of their slopes is -1 .
Three points \(A, B\) and \(C\) are collinear, if and only if slope of \(A B=\) slope of \(B C\).
Equation of the horizontal line having distance \(a\) from the \(x\)-axis is either \(y=a\) or \(y=-a\).
Equation of the vertical line having distance \(b\) from the \(y\)-axis is either \(x=b\) or \(x=-b\).
The point \((x, y)\) lies on the line with slope \(m\) and through the fixed point \(\left(x_\alpha, y_0\right)\), if and only if its coordinates satisfy the equation \(y-y_0=m\left(x-x_0\right)\).
Equation of the line passing through the points \(\left(x_1, y_1\right)\) and \(\left(x_2, y_2\right)\) is given by \( y-y_1=\frac{y_2-y_1}{x_2-x_1}\left(x-x_1\right) . \)
The point \((x, y)\) on the line with slope \(m\) and \(y\)-intercept \(c\) lies on the line if and only if \(y=m x+c\).
If a line with slope \(m\) makes \(x\)-intercept \(d\). Then equation of the line is \(y=m(x-d)\).
Equation of a line making intercepts \(a\) and \(b\) on the \(x\)-and \(y\)-axis, respectively, is \(\frac{x}{a}+\frac{y}{b}=1\).
The equation of the line having normal distance from origin \(p\) and angle between normal and the positive \(x\)-axis \(\omega\) is given by \(x \cos \omega+y \sin \omega=p\).
Any equation of the form \(\mathrm{Ax}+\mathrm{B} y+\mathrm{C}=0\), with \(\mathrm{A}\) and \(\mathrm{B}\) are not zero, simultaneously, is called the general linear equation or general equation of a line.
The perpendicular distance \((d)\) of a line \(\mathrm{A} x+\mathrm{B} y+\mathrm{C}=0\) from a point \(\left(x_1, y_1\right)\) is given by \(d=\frac{\left|\mathrm{A} x_1+\mathrm{B} y_1+\mathrm{C}\right|}{\sqrt{\mathrm{A}^2+\mathrm{B}^2}}\).
Distance between the parallel lines \(\mathrm{A} x+\mathrm{B} y+\mathrm{C}_1=0\) and \(\mathrm{A} x+\mathrm{B} y+\mathrm{C}_2=0\), is given by \(d=\frac{\left|\mathrm{C}_1-\mathrm{C}_2\right|}{\sqrt{\mathrm{A}^2+\mathrm{B}^2}}\).