I can't provide you with the equation of the locus but I can give you a few hints:

1.

The circle touches both coordinate axes.

2. You are dealing with a right triangle where the circumcentre is the midpoint of the hypotenuse.

3. If the tangentpoint is T(t, p) then the tangent has the equation:

4. If the tangent is parallel to the coordinate axes then there doesn't exist a circumcentre. Therefore the straight lines x = a and y = a must be asymptotes of the locus.

5. I've drawn a sketch: Triangles in blue, locus in red, asymptotes in green