Thread: A problem with complex numbers and inequalities

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!

Originally Posted by Dark Sun

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.

What do you mean by smallest and largest, if z is a complex number?

If that may help you: $\displaystyle |z-a|+|z+a|=2c$ is the equation of an ellipse with foci $\displaystyle \pm a$, long half-axis $\displaystyle c$ and short half-axis $\displaystyle \sqrt{c^2-|a|^2}$.

I'm sorry, I meant the largest and smallest values of |z|.

I will look into this ellipse, and report back.

Originally Posted by Dark Sun

I'm sorry, I meant the largest and smallest values of |z|.

I will look into this ellipse, and report back.

Then the extreme values are 0 and $\displaystyle |c|$. Triangular inequality from $\displaystyle 2z=(z+a)+(z-a)$, and considering $\displaystyle z=|c|\frac{a}{|a|}$ should do it.