Determine the complex number z from the conditions…

Determine the complex number $z$ from the conditions $\left|\dfrac{z}{z+1}\right|=1$ and $\dfrac{z}{\bar z}=i$.

Condition 1 · $\left|\dfrac{z}{z+1}\right|=1$

Step 1 · Rewrite as a distance condition

$$ \frac{|z|}{|z+1|}=1 \quad\Longrightarrow\quad |z|=|z+1| $$

Geometrically: the distance from $z$ to $0$ equals the distance from $z$ to $-1$ — the perpendicular bisector of the segment joining $0$ and $-1$.

Step 2 · Solve in coordinates

With $z=x+iy$: $\sqrt{x^2+y^2}=\sqrt{(x+1)^2+y^2}$. Squaring:

$$ x^2+y^2 = (x+1)^2+y^2 = x^2+2x+1+y^2 \quad\Longrightarrow\quad 0=2x+1 \quad\Longrightarrow\quad x=-\frac12 $$

$$ \boxed{z = -\frac12+iy,\quad y\in\mathbb{R}} $$

Condition 2 · $\dfrac{z}{\bar z}=i$

Step 3 · Rewrite and substitute

$$ z = i\bar z $$

With $z=x+iy$, $\bar z=x-iy$:

$$ x+iy = i(x-iy) = ix+y $$

Step 4 · Equate real and imaginary parts

Real: $x=y$; Imaginary: $y=x$ — the same condition:

$$ \boxed{x=y} $$

Combine the two conditions

Step 5 · Intersect

From Condition 1, $x=-\frac12$; from Condition 2, $y=x$. So $y=-\frac12$ as well:

$$ \boxed{z = -\frac12-\frac12 i = -\frac{1+i}{2}} $$

Step 6 · Verify Condition 1

$$ z=-\tfrac12-\tfrac12i, \qquad z+1=\tfrac12-\tfrac12i $$

$$ |z| = \frac{\sqrt2}{2} = |z+1| \quad\Longrightarrow\quad \left|\frac{z}{z+1}\right|=1 \quad \checkmark $$

Step 7 · Verify Condition 2

$$ \frac{z}{\bar z} = \frac{-\frac12-\frac12i}{-\frac12+\frac12i} = \frac{1+i}{1-i} = \frac{(1+i)^2}{2} = \frac{2i}{2} = i \quad \checkmark $$

Alternative route via polar form

Write $z=re^{i\theta}$, so $\bar z=re^{-i\theta}$ and $\dfrac{z}{\bar z}=e^{2i\theta}$. Condition 2 becomes $e^{2i\theta}=e^{i\pi/2}$, i.e. $\theta=\frac{\pi}{4}+\pi k$. Combined with $x=-\frac12<0$ (so $z$ is in the third quadrant), $\theta=\frac{5\pi}{4}$, and matching $x=r\cos\theta=-\frac{r\sqrt2}{2}=-\frac12$ gives $r=\frac{\sqrt2}{2}$, reproducing $z=-\frac12-\frac12i$.