JEE Main & Advanced Mathematics Straight Line Angle Between two Non-parallel Lines

If \[\theta \]  be the angle between the lines \[y={{m}_{1}}x+{{c}_{1}}\] and \[y={{m}_{2}}x+{{c}_{2}}\] and intersecting at A. Then, \[\theta ={{\tan }^{-1}}\,\left| \frac{{{m}_{1}}-{{m}_{2}}}{1+{{m}_{1}}{{m}_{2}}} \right|\].   If \[\theta \] is angle between two lines, then \[\pi -\theta \] is also the angle between them.

(1) Angle between two straight lines when their equations are given : The angle q between the lines \[{{a}_{1}}x+{{b}_{1}}y+{{c}_{1}}=0\] and

\[{{a}_{2}}x+{{b}_{2}}y+{{c}_{2}}=0\] is given by, \[\tan \theta =\left| \,\frac{{{a}_{2}}{{b}_{1}}-{{a}_{1}}{{b}_{2}}}{{{a}_{1}}{{a}_{2}}+{{b}_{1}}{{b}_{2}}}\, \right|\].

(2) Conditions for two lines to be coincident, parallel, perpendicular and intersecting : Two lines \[{{a}_{1}}x+{{b}_{1}}y+{{c}_{1}}=0\]  and \[{{a}_{2}}x+{{b}_{2}}y+{{c}_{2}}=0\] are,

(a) Coincident, if  \[\frac{{{a}_{1}}}{{{a}_{2}}}=\frac{{{b}_{1}}}{{{b}_{2}}}=\frac{{{c}_{1}}}{{{c}_{2}}}\]             

(b) Parallel, if \[\frac{{{a}_{1}}}{{{a}_{2}}}=\frac{{{b}_{1}}}{{{b}_{2}}}\ne \frac{{{c}_{1}}}{{{c}_{2}}}\]

(c) Intersecting, if \[\frac{{{a}_{1}}}{{{a}_{2}}}\ne \frac{{{b}_{1}}}{{{b}_{2}}}\]

(d) Perpendicular, if  \[{{a}_{1}}{{a}_{2}}+{{b}_{1}}{{b}_{2}}=0\]