Hello, prasum!
I'm not sure I understand the problem . . .
Find the locus of centres of circle: $\displaystyle x^2+y^22ax2by+2\:=\:0$
. . where $\displaystyle a$ and $\displaystyle b$ are parameters,
if the tangents from any point to each of the circles are orthogonal.
The circle has the equation: .$\displaystyle (xa)^2 + (yb)^2 \:=\:a^2+b^22 $
It has center $\displaystyle C(a,b)$ and radius $\displaystyle r\,=\,\sqrt{a^2+b^22}$
Let $\displaystyle P(h,k)$ be any point (exterior to the circle).
Tangents are drawn from $\displaystyle P$ to $\displaystyle A$ and $\displaystyle B$ on the circle.
. . The tangents are orthogonal: .$\displaystyle \angle P = 90^o$
Code:

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

Since $\displaystyle \angle A = \angle B = \angle P = 90^o,\;PA = PB,\:CA = CB$
. . then quadrilateral $\displaystyle APBC$ is a square.
Its diagonal is: .$\displaystyle CP \,=\,\sqrt{2}\,r \;=\;\sqrt{2(a^2+b^22)}$
Therefore, the locus of the centers of the circles
. . is a circle with center $\displaystyle (h,k)$ and radius $\displaystyle \sqrt{2(a^2+b^22)}$