Equation (a(b-c)x^2+b(c-a)x+c(a-b)=0) has equal roots.
Discriminant = 0: [ [b(c-a)]^2-4\cdot a(b-c)\cdot c(a-b)=0 ]
Expanding and simplifying yields (2/b=1/a+1/c), i.e. (a,b,c) in H.P.
Proved.
11TH•Mathematics•Sequences And Series•Question 18
Overview
If the roots of the equation $a(b-c)x^2 + b(c-a)x + c(a-b) = 0$ are equal, prove that $a,\ b,\ c$ are in H.P.
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