If Z is a complex number such that |z| greater than or equal to 2, then the minimum value of ∣∣z+12∣∣.
|z|=2 is a circle with centre as origin and radius 2. |z|≥2 ⇒ all the points outside and on the circle
|z| =2.
It is the shaded region in the figure.
The closest point in that region on the negative x-axis, which is (-2,0). So the least distance is −12-(-2)= 32.
Or
Another way of solving is by applying triangle inequality.
∣∣z+12∣∣≤|z|−∣∣12∣∣
=2-∣∣12∣∣
= 32