JEE Main & Advanced Mathematics Straight Line Length of Perpendicular

(1) Distance of a point from a line : The length p of the perpendicular from the point \[({{x}_{1}},\,{{y}_{1}})\] to the line \[ax+by+c=0\] is given by \[p=\frac{|a{{x}_{1}}+b{{y}_{1}}+c|}{\sqrt{{{a}^{2}}+{{b}^{2}}}}\].

• Length of perpendicular from origin to the line \[ax+by+c=0\] is \[\left| \,\frac{c}{\sqrt{{{a}^{2}}+{{b}^{2}}}}\, \right|\,\].
• Length of perpendicular from the point \[({{x}_{1}},\,{{y}_{1}})\] to the line \[x\cos \alpha +y\sin \alpha =p\] is \[|{{x}_{1}}\cos \alpha +{{y}_{1}}\sin \alpha -p|\].

(2) Distance between two parallel lines : Let the two parallel lines be \[ax+by+{{c}_{1}}=0\] and \[ax+by+{{c}_{2}}=0\].

First Method: The distance between the lines is \[d=\frac{|{{c}_{1}}-{{c}_{2}}|}{\sqrt{({{a}^{2}}+{{b}^{2}})}}\].

Second Method: The distance between the lines is \[d=\frac{\lambda }{\sqrt{({{a}^{2}}+{{b}^{2}})}}\],

where (i) \[\lambda =|{{c}_{1}}-{{c}_{2}}|\], if they be on the same side of origin.

(ii) \[\lambda =|{{c}_{1}}|+|{{c}_{2}}|\], if the origin O lies between them.

Third method : Find the coordinates of any point on one of the given line, preferably putting \[x=0\] or \[y=0\]. Then the perpendicular distance of this point from the other line is the required distance between the lines.

Distance between two parallel lines \[ax+by+{{c}_{1}}=0\],\[kax+kby+{{c}_{2}}=0\] is \[\frac{\left| \,{{c}_{1}}-\frac{{{c}_{2}}}{k}\, \right|}{\sqrt{{{a}^{2}}+{{b}^{2}}}}\]. Distance between two non parallel lines is always zero.