If the roots of the equation $a(b-c)x^2+b(c-a)x+c(a-b)=0$ has equal roots , then a, b, c are in 

S = Sum of the co-efficients 

= a(b - c) + b(c - a)+ c(a - b) 

= ab - ac + bc - b a + ca - ab = 0 

Thus, 1 is a root Now, product of the roots= 1. 1 = 1

$\begin{array}{l}\frac{c(a-b)}{a(b-c)}=1\\ ac-bc=ab-ac\\ 2ac=ab+bc\\ b(a+c)=2ac\\ b=\frac{2ac}{(a+c)}\end{array}$

Thus, a, b and c are in H.P