Let z and w be two complex numbers, such that w=z z-2z+2, |z+iz-3i|=1 and Re(w) has minimu…

Question

Let $z$ and $w$ be two complex numbers, such that $w=z\bar z-2z+2$, $\left|\dfrac{z+i}{z-3i}\right|=1$ and $\operatorname{Re}(w)$ has minimum value. Then the minimum value of $n\in\mathbb N$ for which $w^n$ is real, is equal to

Answer: 4

Accepted Answer

$|z+i|=|z-3i|$ means $z$ lies on the perpendicular bisector of $(0,-1)$ and $(0,3)$, i.e. $y=1$.

$w=|z|^2-2z+2=(x^2+y^2-2x+2)+i(-2y)$. With $y=1$: $w=(x-1)^2+2-2i$, minimized at $x=1$, giving $w=2-2i=2\sqrt2 e^{-i\pi/4}$.

$w^n=(2\sqrt2)^n e^{-in\pi/4}$ is real when $\sin\dfrac{n\pi}4=0$, i.e. $n$ a multiple of 4. Minimum such $n\in\mathbb N$ is 4.

Solutions · 1