dude for the second question you can assume the two tangents as y=mx+c and y=nx+d we can calulate c and d by condition of tangency i don't think sign of c or d should matter now we know a relation between m and n as angle between two tangents is 60 both these equations satisfy the point of intersection now eliminating m and n will be painfull job i think the locus will be a straight line please correct me if i am wrong