<s>[INST] Plot $|z - i| + |z + i| = 16$ on the complex plane 
Plot $|z - i| + |z + i| = 16$ on the complex plane

Conceptually I can see what is going on. I am going to be drawing the set of points who's combine distance between $i$ and $-i = 16$, which will form an ellipse. I was having trouble getting the equation of the ellipse algebraically.
I get to the point:
$x^2 + (y - 1) ^2 + 2\sqrt{x^2 + (y - 1)^2}\sqrt{x^2 + (y + 1)^2} + x^2 + (y+ 1)^2 = 256$
$2x^2 + 2y^2 + 2\sqrt{x^2 + (y - 1)^2}\sqrt{x^2 + (y + 1)^2} = 254$
It seems like I'm and doing something the hard way.
 [/INST] Maybe it's quicker way.
Equation $|z-i| + |z+i| = 16$ is equivalent to
$$
\sqrt{x^2 + (y-1)^2} = 16 - \sqrt{x^2 + (y+1)^2}.
$$
Squaring both sides you obtain
$$
x^2 + (y-1)^2 = 256 - 32\sqrt{x^2+(y+1)^2} + x^2 + (y+1)^2.
$$
Some terms cancel out hence you get
$$
8\sqrt{x^2 + (y+1)^2} = 64 + y,
$$
and
$$
64(x^2 + (y+1)^2) = 64^2 + 128y + y^2.
$$
Finally
$$
64x^2 + 63y^2 = 64 \cdot 63.
$$
Now it is easy to write ellipse equation
$$
\frac{x^2}{63} + \frac{y^2}{64} = 1.
$$
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