A circle passes through the point $A,$ and is tangent to the $y$-axis at the point $B,$ as shown. Find the radius of the circle.

In the diagram, A = (1,0) and B = (0,1).

If you construct a unit circle at a point, say (1,1), you will see that it satisfies the given constructions.

The radius of a unit circle is just 1 unit.