So it's a hyperbola when

?

Mr F says: No. There's no locus because there's no value of z that satisfies the condition. If you take an algebraic approach by substituting z = x + iy into |z - 1| - |z + 1| = 4, say, you might be fooled into thinking that yuo get an ellipse. However, there is an important restriction that arises during the calculation. When you impose this restriction, you end up with the null set.
I guess it would be a parabola when

? Mr F says:

No. It's a ray, the degenerate case of the above ....
And if

it's a ellipse?

Mr F says: No. It's a single branch of a hyperbola. Which branch will depend on whether c is positive or negative.
And I must find the equation of this hyperbola?

something to do with

and

?

is the length from the foci to the center and

is the length from the vertex to the center. What would this be centered at though? The origin?

Darn, I wish I remembered the formula for a hyperbola. I am sure

and

are used somehow.