A(a,b); C(c,d)
O(0,0)
to find if AO=CO or if AO^2 = CO^2 or if a^2 + b^2 = c^2 + d^2 ---eqn1
a)a/b = c/d
a can be equal to c and then b will be equal to d and OA will be equal to OC
but another case: a=3c;b=3d; OA=3OC
Insufficient
b)|a| + |b| = |c| + |d|
squaring both sides
a^2 + b^2 + 2|ab| = c^2 + d^2 + 2|cd| ---eqn2
we don't know whether |ab|=|cd|
insufficient

a&b together)
a/b=c/d = x
a=bx;c=dx
|bx| + |b| = |dx| + |d|
|b| = |d|
|b*b| = |d*d|
|b*b*x| = |d*d*x|
|ba| = |dc|
|ab| = |cd|
thus from eqn2,
a^2 + b^2 = c^2 + d^2
thus from eqn1, OA=OC
Sufficient
IMO C