Derive the condition for a line to be tangent to a circle

Question

Given the circle $(x-a)^2+(y-b)^2 = r^2$ and the line, $ux+vy+w = 0$, show that the condition for the line to be tangent to the circle is $(w+vb+ua)^2 =r^2(v^2+u^2) $.

I have a solution but, as usual, it is messy and I am hoping for a nicer one.

I'll post mine in a few days if no one has a better one.

Note: When I entered this question, a number of similar questions were suggested, but none of them ask precisely this question.

Answer 1

This is straightforward: the distance of the centre from the line should be $r$, so$$\frac{|ua+vb+w|}{\sqrt{u^2+v^2}}=r$$