Question

A line is such that the algebraic sum of perpendicular in it fro a number of point is zero. Prove that the line always passes through a fixed point.

Answer 1

$lx+my-1=0s.t.\stackrel{n}{\underset{i=1}{\Sigma }}\frac{({lx}_i+m{y}_i-1)}{\sqrt{(l^2+m^2)}}=0$
$\therefore l\Sigma x_i+m\Sigma y_i-n=0$.
or $\frac{l\Sigma x_i}{n}+\frac{m\Sigma y_i}{n}-1=0$
Above shows that the line $lx+my-1=0$ passes through the centroid $(\frac{\Sigma x_i}{n},\frac{\Sigma y_i}{n})$ of the given points.