Question

If the roots of equation (c^2-ab)x^2-2(a^2-bc)x b^2-ac=0 in x are equal then show that either a=0 or a^3 b^3 c^3=3abc

Answer 1

c2 – ab)x2 – 2(a2 – bc)x + b2 – ac = 0 has equal roots.
So, D = 0
4(a2 – bc)2 – 4(c2 – ab)(b2 – ac) = 0
(a4 + b2c2 – 2a2bc) – (b2c2- ab3 – ac3 + a2bc) = 0
a4 + ab3 + ac3 – 3a2bc = 0
a(a3 + b3 + c3 – 3abc) = 0
so, either a = 0 or (a3 + b3 + c3 – 3abc) = 0