<s>[INST] Need help with complex equation: |z−i|+|z+i|=2 I am trying to solve this equation: |z−i|+|z+i|=2 and don't know how to do it. This what I have:
$$\sqrt{(x+1)^2+y^2}+ \sqrt{(x-1)^2+y^2} = 2 /^2$$
$$(x+1)^2+y^2+(x-1)^2+y^2 + 2\sqrt{[(x+1)^2+y^2][(x-1)^2+y^2]} = 4$$
$$x^2 +2x + 1+y^2+x^2-2x + 1+y^2 + 2\sqrt{[(x^2 +2x + 1)+y^2][(x^2-2x + 1)+y^2]}=4$$
$$2x^2+2y^2 + 2 + 2\sqrt{[(x^2 +2x + 1)+y^2][(x^2-2x + 1)+y^2]}=4$$
$$2x^2+2y^2 + 2 + 2\sqrt{(x^2 +2x + 1)(x^2-2x + 1)+x^2y^2-2xy^2+y^2+x^2y^2-2xy^2+y^2+y^4}=4$$
$$2x^2+2y^2 + 2 + 2\sqrt{(x^2 +2x + 1)(x^2-2x + 1)+2x^2y^2-4xy^2+2y^2+y^4}=4$$
$$2x^2+2y^2 + 2 + 2\sqrt{x^4+2x^3+x^2-2x^3+4x^2-2x+x^2+2x+1+2x^2y^2-4xy^2+2y^2+y^4}=4$$
$$2x^2+2y^2 + 2 + 2\sqrt{x^4+6x^2+1+2x^2y^2-4xy^2+2y^2+y^4}=4$$
And I have no idea what to do next.
Any help would be much appreciated
 [/INST] Triangle inequality tells us $|z+i| + |z-i| \ge |(z+i)-(z-i)| = |2i| = 2$ with equality holding if and only if geometrically of $0$ is colinear and between $z+i$ and $z-i$.
The only values colinear between $z + i$ and $z-i$ must all have the same real value so the real part as $z$ has and if $0$ is one such number, the real part of $z$ must be $0$.
If $Im(z) = b$ then $z = bi$ and we have $|bi+i| + |bi - i| = |b+1|+|b-1|=2$.  By cases we can verify then this will hold if and only if $b \in [-1,1]$.
(If $b > 1$ then $|b+1| > 0$ and $|b-1| > 2$ and if $b < -1$ then $|b+1| > 2$ and $|b-1| > 0$; but if $-1 < b < 1$ then $|b+1| = b+1$ and $|b-1|= 1-b$ and $|b+1|+|b-1| = (b+1)+(1-b)=2$.)
So $z$ can be any $bi; -1 \le b \le 1$.
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Alternatively (in hindsight) note:
If $z = a + bi$ then $|z\pm i| = \sqrt{a^2 + (b\pm 1)^2}$ so
$|z+i| +|z-i|$
$=\sqrt {a^2 + (b+ 1)^2} + \sqrt {a^2 + (b-1)^2}$
$[*]\ge\sqrt {(b+1)^2} +\sqrt {(b-1)^2}$
$ =|b+1|+|b-1|$
$=\begin{cases}(b+1)+(b-1)=2b>2& b> 1\\ (b+1)+(1-b)=2 &-1\le b \le 1\\(-b-1)+(1-b)=-2b>2 & b < -1\end{cases}$
$[**] \ge 2$
with equality holding if and only if $a = 0$ and $-1 \le b \le 1$.
[*] Equality holding if and only if $a = 0$.
[**] Equality holding if and only if $-1 \le b \le 1$.
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