Set of points $\left| \frac{z+i\bar z}{2}\right|=\left| z-\frac{1+i}{2}\right|$

Question

Find the set $P$ of points $M(z)$ verifying $\left| \dfrac{z+i\bar z}{2}\right|=\left| z-\dfrac{1+i}{2}\right|$

I think it's a parabola but I can't find its equation. Please include the steps to this problem.

Answer 1

Let $z = x+iy \therefore \bar{z} = x-iy$ The desired locus is $$\sqrt{(x+y)^2+(x+y)^2} = 2\sqrt{(x-\frac{1}{2})^2+(y-\frac{1}{2})^2}$$ The given locus is the set of all points whose distance from the point $(\frac{1}{2},\frac{1}{2})$ is $\frac{1}{\sqrt2}$ times the distance from the line $x+y=0$ which is an ellipse with the point as focus and line as directrix.

Answer 2

$\displaystyle \left |\frac{x+iy+i(x-iy)}{2} \right |=\left | \frac{2x+2iy-1-i}{2} \right |$ $$\left |\frac{(x+y)+i(y+x)}{2} \right |=\left | \frac{(2x-1)+i(2y-1)}{2} \right |$$ $$\sqrt{(x+y)^2+(y+x)^2}=\sqrt{(2x-1)^2+(2y-1)^2}$$ $$x^2+y^2+2xy+y^2+x^2+2xy=4x^2+1-4x+4y^2+1-4y$$ $$2x^2+2y^2+4xy=4x^2+4y^2+2-4x-4y$$ $$2x^2+2y^2-4xy-4x-4y+2=0$$ $\Rightarrow$ $$x^2+y^2-2xy-2x-2y+1=0$$