Notice the typo: the first = should have been – (otherwise the question doesn't make sense).

The difference between the arguments of two complex numbers is the argument of their quotient. So . You want that argument to be .

Now a complex number has argument if (and ).

So let . Then . Find the real and imaginary parts of that number in the usual way (by rationalising the denominator) and write down the condition that the real part is the negative of the imaginary part. That will give you the equation of a circle. If you want extra credit, you could point out that the locus is in fact only part of that circle (because of the condition in the previous paragraph, which says that the imaginary part of should be positive).

If (constant) then . Putting again, that becomes , which is the equation of a circle. Compare it with the equation of the circle that you got in the first part of the question, and see if there is a value of k that makes those two equations the same.