2010/006) equation sin(w) = (a^2 + b^2 + c^2)/(ab+bc+ca), where a,b,c are fixed nozero real numbers

has a solution for w
A) Whatever be a,b,c;
B) iff a^2 + b^2 + c^2 < 1;
C) iff a,b and c all lie in the interval (-1,1);
D) iff a = b = c;
Explain your answer.

ans

we know by GM AM enaquality a^2+b^2 >= 2ab
a^2+c^2 >= 2ac
and b^2+c^2 >= 2bc

adding all 3 we get 2(a^2+b^2+c^2) >= 2(ab+bc+ca)

or (a^2+b^2+c^2)/(ab+bc+ca)>= 1 and equal only when a= b= c

sin (w) cannot be > 1 so ans is when (a^2+b^2+c^2)/(ab+bc+ca) = 1 or a= b =c

that is d