Find the values of non-negative real numbers h1,h2,h3,k1,k2,k3 such that the algebraic sum of the perpendiculars drawn from points (2,k1),(3,k2),(7,k3),(h1,4),(h2,5),(h3,−3) on a variable line passing through (2,1) is zero.

Let the equation of variable line be ax+by+c=0, it is given that ∑i=16axi+byi+ca2+b2=0⇒a(∑xi6)+b(∑yi6)+c=0So, the fixed point must be ∑xi6, ∑yi6. But fixed point is (2,1) so 2+3+7+h1+h2+h3/6=2⇒h1+h2+h3=0⇒h1=0, h2=0, h3=0(as h1, h2, h3 are non-negative) similarly, we getk1+k2+k3+4+5+−36=1⇒k1=k2=k3=0