If p,x1,x2....  and q,y1,y2...  are in A.P with common difference a  and b  respectively, then the centre of mean position of the points Ai(xi,yi),i=1,2,3,...n  lies on the line

Let the coordinates of the centre of mean position of the pointsAi,i=1,2,3,.....n be (x,y) thenx=xi+x2+...+xnn,y=y1+y2+...+ynnx=p+a+p+2a+...+p+nan=np+a(1+2+....+n)n  =p+n(n+1)2na=p+n+12a     ⇒2x-pa=n+1→1similarly ,y=nq+b(1+2+...+n)n=q+n(n+1)2nb=q+n+12b ⇒2y-qb=n+1→2 from 1,2   ⇒2(x−p)a=2(y−q)b⇒bx−ay=bp−aq