Hello, prasum!
I'm not sure I understand the problem . . .
The circle has the equation: .
It has center and radius
Let be any point (exterior to the circle).
Tangents are drawn from to and on the circle.
. . The tangents are orthogonal: .
Code:

 * * *
 * * A
 * ♥
 * o *o
 o o
 * C o * o P
 * (a,b)♥ * ♥(h,k)
 * o * o
 o o
 * o *o
 * ♥
 * * B
 * * *
  +                     

Since
. . then quadrilateral is a square.
Its diagonal is: .
Therefore, the locus of the centers of the circles
. . is a circle with center and radius