Set of complex numbers that satisfy a given equation

The equation is $\displaystyle |z-a| - |z+a| = 2c$

i) Let $\displaystyle a \in \mathbb{R}$ and $\displaystyle c > 0$ be fixed. Describe the set of points $\displaystyle z$ satisfying the above equation for every possible choice of $\displaystyle a$ and $\displaystyle c$.

ii) Now let $\displaystyle a \in \mathbb{C}$ 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 $\displaystyle z$ 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!