The correct options are
A arg(−1−i)=π4, where i=√−1
B The function f:R→(−π,π], defined by f(t)=arg(−1+it) for all t∈R, is continuous at all points of R, where i=√−1
D For any three given distinct complex numbers z1,z2 and z3, the locus of the point z satisfying the condition arg((z−z1)(z2−z3)(z−z3)(z2−z1))=π,
lies on a straight line.
arg(−1−i)=π4−π=−3π4
f(t)=arg(−1+it)={π−tan−1t, t≥0−tan−1t−π, t<0
Clearly, f is discontinuous at t=0.
arg(z1z2)−arg(z1)+arg(z2)
=arg(z1)−arg(z2)+2nπ−arg(z1)+arg(z2)
=2nπ
arg((z−z1)(z2−z3)(z−z3)(z2−z1))=π
⇒arg(z−z1z2−z1)+arg(z2−z3z−z3)=π
⇒z, z1, z2, z3 are concyclic.
Hence, locus of z is a circle.