complex no.

The locus of z is a straight line. The distance of i and 1 from the line are equal.
If z = x + yi then x^2 + (y - 1)^2 = (x - 1)^2 + y^2 => x = y
The locus of w is a circle of radius 1 and centre (2,0).
P and Q is closest when the line joining PQ is perpendicular to z and passes through the centre of w.
Distance of P from centre of w is | 2 - 0 | / √(1 + 1) = √2
Therefore PQ = √2 - radius = √2 - 1

圖片參考：http://img80.imageshack.us/img80/9077/loci.png