Set of complex numbers that satisfy a given equation

The equation is

i) Let and be fixed. Describe the set of points satisfying the above equation for every possible choice of and .

ii) Now let and, using a rotation of the plane, describe the locus of points satisfying the above equation.

I tried algebra for both i) and ii) and just got a big mess, no matter what form I converted the complex number to, or even just matching up real and imaginary parts.

I am not sure how else to find the set of that would satisfy the above equation.

I am also lost by what they mean by locus in ii), I am assuming it must be some sort of conic section, either a hyperbola or an ellipse. But I dont know how to show this either.

Any advice, nudge or hint in the right direction would be helpful. Thanks for answering!