Question

Describe the set of points z in the complex plane that satisfy the given equation; |z-2|=Re(z).

Answer 1

Parabola with vertex at point (1,0) that goes to the right (see attached diagram).

Explanation:

Let the complex number z be

`z=x+iy,`

then

`z-2=x+iy-2=(x-2)+iy\\ \\Re\ (z-2)=x-2\\ \\Im\ (z-2)=y\\ \\|z-2|=√(Re^2\ (z-2)+Im^2\ (z-2))=√((x-2)^2+y^2)`

and

`Re\ z=x`

Thus,

`√((x-2)^2+y^2)=x`

Square it:

`(x-2)^2+y^2=x^2\\ \\x^2-4x+4+y^2=x^2\\ \\-4x+4+y^2=0\\ \\y^2=4x-4\\ \\y^2=4(x-1)`

This is the equation of parabola with vertex at point (1,0) that goes to the right.