For any complex number $z$, the minimum value of $|z|+|z-1|$ is
For any complex number $z$, the minimum value of $|z|+|z-1|$ is
1
0
$1 / 2$
$3 / 2$
Solution
Let $z=x+i y$
$
\begin{aligned}
|z| & =\sqrt{x^2+y^2} \\
|z-1| & =\sqrt{(x-1)^2+y^2}=\sqrt{x^2-2 x+1+y^2}
\end{aligned}
$
For minimum value of $|z|$ and $|z-1|$
$
\begin{aligned}
& |z-1| \geq 0 \\
& x-1 \geq 0 \left[\because y^2 \geq 0\right]\\
& x \geq 1
\end{aligned}
$
The minimum value of $|z|+|z-1|$ is 1