W.t.s. there are complex numbers z satisfying |z-a|+|z+a|=2|c| iff $\displaystyle |a|\leq |c|.$ Also, I need to find the smallest and largest values of z provided this condition is fulfilled.

Step 1 is simple: Assume there are complex numbers z satisfying |z-a|+|z+a|=2|c|. Then, $\displaystyle 2|c|=|z-a|+|z+a|=|a-z|+|z+a|\geq 2|a|,\mbox{ hence }|c|\geq |a|.$

So far for step 2 I have: Assume that the condition $\displaystyle |a|\leq |c|$ is fullfilled. If z=-|c| is our smallest value, and z=|c| is our largest value, then we have $\displaystyle 2|c|=2|z|\leq |z-a|+|z+a|$, and I am now lacking equality.

If I could show that $\displaystyle \frac{|c|-a}{|c|+a}\geq 0$, this would provide equality, however, there is still the matter of the complex number a, and since there is no order relation for complex numbers, this is proving difficult.

I have also gotten $\displaystyle |z-a|+|z+a|\leq |z|+|-a|+|z|+|a|=2|c|+2|a|\leq 4|c|$, which is off by a factor of 2.

I think my value for z is wrong.

Any help would be greatly appreciated!