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> **Question:** Let $u = 3 - i$ and $v = 2 + i$ and consider the locus $|z - v| = 2$. Find the minimum value of $|z - u|$ where $z$ is any point on the locus.
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What I have done;

First of all by using

$$ |z-v| = 2 $$

I got a circle centered at $(2 , i)$ with a radius $2$

and by using 

$$ |z-u| $$

I have a point located at $(3 , -1)$

After this I have found a straight line that passes through these 2 points with the equation

$$ y = -2x + 5 $$

This line intersects the circle at two points

Shown here 

[![enter image description here](https://i.stack.imgur.com/XXqjz.png)](https://i.stack.imgur.com/XXqjz.png)

So how do I find the 'minimum' value of $ |z-u| $ , I have to show most things algebraically (as a side note..)