The algebraic sum of the perpendicular distances from A(a1,b1), B(a2,b2) and C(a3,b3) to a variable line is zero, then the line passes through
(a) the orthocentre of ΔABC
(b) the centroid of ΔABC
(c) the circumcentre of ΔABC
(d) none of these
Let the variable line be y=mx+c or mx-y+c=0.
Sum of algebraic distances = \(\frac {mx_1-y_1+c}{\sqrt {m^2+1}}+\frac {mx_2-y_2+c}{\sqrt {m^2+1}}+\frac {mx_3-y_3+c}{\sqrt {m^2+1}}=0\)
(Mod is not to be used when we imply algebraic distances.)
\(\Rightarrow m(x_1+x_2+x_3)-(y_1+y_2+y_3)+3c=0\)
\(\Rightarrow \frac {y_1+y_2+y_3}{3}=m\frac {x_1+x_2+x_3}{3}+c\)
Hence, centroid.
Thank You Sir.