The algebraic sum of perpendicular distances from `A(x_(1),y_(1)), B(x_(2),y_(2))` and `C(x_(3),y_(3))` to a variable line is zero, then all the such lines will always pass thorugh

Options

A. the orthocentre of `Delta ABC`
B. the centroid of `Delta ABC `
C. the circumcentre of `DeltaABC`
D. none of these

Answer

B

Explanation

Let `Px + Qy = 1` be a variable line, where P,Q are variable. By hypothesis , we have .
`(Px_(1)+Qy_(1)-1)/(sqrt(p^(2) +Q^(2)) +(p^(2)+Q^(2)))+(Px_(3)+Qy_(3)-1)/(sqrt(P^(2)+Q^(2)))=0`
`rArr P(x_(1) + x_(2) + x_(3)) +Q(y_(1) + y_(2)+y_(3)) = 3`
`rArr p((x_(1)+x_(2)+x_(3))/(3))+Q((y_(1)+y_(2)+y_(3))/(3))=1`
`rArr ((x_(1) + x_(2)+x_(3))/(3),(y_(1)+y_(2)+y_(3))/(3))` lies on `Px + Qy=1`
Hence, the variable line passig thorought the the centriod of the triangle ABC.