Question

If z=x+iy and |z-1|^2+|z+1|^2=4, determine the position of the points z in the complex plane?

`|z-1|^2+|z+1|^2=4`

Answer 1

Please see below.

Explanation:

As `z=x+iy` we can write `|z-1|^2+|z+1|^2=4` as

`|x-1+iy|^2+|x+1+iy|^2=4`

or `(x-1)^2+y^2+(x-1)^2+y^2=4`

or `2x^2+2+y^2=4`

or `2x^2+y^2=2`

or `x^2/1+y^2/2=1`

Hence,the position of the points `z` in the complex plane is given by the ellipse `x^2/1+y^2/2=1`, which is displayed below.

graph{x^2/1+y^2/2=1 [-4, 4, -2, 2]}