(a) 在同一个 Argand 图中，画出由方程 $|z-3+2\mathrm{i}|=2$ 和 $|w-3+2\mathrm{i}|=|w+3-4\mathrm{i}|$ 给出的轨迹草图，其中 $z$ 和 $w$ 为复数。

(b) 由此求这些轨迹上的点所对应的 $|z-w|$ 的最小值。答案以精确形式表示。

Answer all questions. You should use a calculator where appropriate.

You must show all necessary working clearly; no marks will be given for unsupported answers from a calculator.

Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place for angles in degrees, unless a different level of accuracy is specified in the question.

请回答所有题目。适当时应使用计算器。必须清楚展示所有必要的计算过程；仅给出计算器答案而没有过程，不予评分。除非题目另有精度要求，非精确数值答案保留 3 位有效数字，以度为单位的角度保留 1 位小数。

第一个轨迹是圆，第二个是到两定点等距的垂直平分线。先找中点及连线斜率。

The locus of $z$ is the circle with centre $A=(3,-2)$ and radius $2$.

For the locus of $w$, the equation means equal distances from $A=(3,-2)$ and $B=(-3,4)$. The midpoint of $AB$ is $(0,1)$, and $AB$ has gradient $-1$.

Its perpendicular bisector therefore has gradient $1$ and equation

Draw this line and the circle on the same Argand diagram, with the centres, reference points and scale clearly marked.

最短距离沿圆心到直线的垂线取得，等于圆心到线的距离减半径。

The midpoint $(0,1)$ of $AB$ lies on the perpendicular bisector, and the line from $A$ to this midpoint is perpendicular to that bisector.

Thus the distance from the circle centre to the line is

Since this exceeds the radius $2$, the least distance between the loci is