If a, b and c are the sides of a A B C such that x^2-2(a+b+c) x+3 (a b+b c+c a)=0 has real roots, then

Step-by-step solution

Since, roots are real, D 0 array ll & 4(a+b+c)^2-12 (a b+b c+c a) 0 & (a+b+c)^2 3 (a b+b c+c a) & a^2+b^2+c^2 (a b+b c+c a)(3 -2) & 3 -2 a^2+b^2+c^2 a b+b c+c a where, a^2+b^2 2 a b, b^2+c^2 2 b c and c^2+a^2 2 c a & a^2+b^2+c^2 (a b+b c+c a) & a^2+b^2+c^2 a b+b c+c a 1 array Also, A= b^2+c^2-a^2 2 b c < 1 b^2+c^2-a^2 < 2 b c Similarly, c^2+a^2-b^2 < 2 c a and a^2+b^2-c^2 < 2 a b a^2+b^2+c^2 < 2(a b+b c+c a) or a^2+b^2+c^2 a b+b c+c a < 2 From Eqs. (i), (ii) and (iii) array lrl & 3 -2 & < 2 & < 4 3 array