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.

Don't we all love algebraic solutions...
The ellipse has an equation of the form
$$(\frac {x} {a})^2 + (\frac {y} {b})^2 = 1$$
The focals are aligned on the y axis, therefore


*

*a is the side of a rectangle triangle, the other site being 1 and the hypothenuse 16/2:


$$a = \sqrt{8^2-1}$$


*

*b is 16/2.


The equation of the ellipse is
$$(\frac {x} {\sqrt{63}})^2 + (\frac {y} {8})^2 = 1$$
Or, to write it as above
$$\frac {x^2} {63} + \frac {y^2} {64} = 1$$
