W.t.s. there are complex numbers z satisfying |z-a|+|z+a|=2|c| iff 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,
So far for step 2 I have: Assume that the condition is fullfilled. If z=-|c| is our smallest value, and z=|c| is our largest value, then we have , and I am now lacking equality.
If I could show that , 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 , which is off by a factor of 2.
I think my value for z is wrong.
Any help would be greatly appreciated!