Suppose A,B,C are defined as A=a2b+ab2−a2c−ac2, B=b2c+bc2−a2b−ab2 and C=a2c+ac2−b2c−bc2, where a>b>c>0 and the equation Ax2+Bx+C=0 has equal roots, then a, b, c are in

A=a(b−c)(a+b+c)B=b(c−a)(a+b+c)C=c(a−b)(a+b+c)Now, Ax2+Bx+C=0⇒ (a+b+c)a(b−c)x2+b(c−a)x+c(a−b)=0Given that roots are equal. Hence, D = 0or  b2c2−2ab2c+b2a2−4a2bc+4acb2+4a2c2−4abc2=0or  b2c2+b2a2+4a2c2+2ab2c−4a2bc−4abc2=0or  (bc+ab−2ac)2=0or  bc+ab=2ac⇒ 1a+1c=2bHence, a, b, c are in H.P.