If (a,b) and (c,d) have to be equidistant from the origin, a^2 + b^2 = c^2 + d^2.
Consider first (1) alone.
Let a = 2, b = 3, c = 4 and d = 6.
Here a/b = c/d = 2/3, but a^2 + b^2 = 13 and c^2 + b^2 = 52.
So (a,b) and (c,d) are not equidistant from the centre.
Next let a = 2, b = 3, c = -2 and d = -3.
Here a/b = c/d = 2/3 and also a^2 + b^2 = c^2 + d^2 = 13.
So (a,b) and (c,d) are equidistant from the centre.
Since nothing definite can be said, (1) alone is not sufficient.
Next consider (2) alone.
It means lal+lbl =lcl+ldl.
Let a = 2, b = 8, c = 5 and d = 5.
Here lal+lbl =lcl+ldl = 10, but a^2 + b^2 = 68 and c^2 + d^2 = 50.
So (a,b) and (c,d) are not equidistant from the centre.
Next let a = 2, b = 8, c = 8 and d = 2.
Here lal+lbl = lcl+ldl = 10 and a^2 + b^2 = c^2 + d^2 = 68.
So (a,b) and (c,d) are equidistant from the centre.
Again nothing definite can be said and so (2) alone is not sufficient.
Next combine both the statements together and check.
Let a/b = c/d = k.
So a = kb and c = kd.
Also from (2), lkbl+lbl = lkdl+ldl.
So (lkl + 1)(lbl - ldl) =0.
Hence either lbl = ldl or lkl = -1 (not possible).
So lbl = ldl, and lal = lkbl =lkdl = lcl.
In this case a^2 + b^2 = c^2 + d^2.
So we can say that both points are equidistant from the centre.
The correct answer is (C).