$\begin{array}{1 1}(A)\;2ax-2by-(a^2+b^2+4)=0 \\(B)\;2ax-2by-(a^2+b^2+4)=0 \\(C)\;2ax-2by+(a^2+b^2+4)=0 \\(D)\;2ax+2by+(a^2+b^2+4)=0 \end{array}$

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Let the variable of the circle is :

$x^2+y^2+2gx+2fg+c=0$

It passes through (x,y)

$\therefore \;a^2+b^2+2gx+2fy+c=0$----(i)

It passes through (a,b)

$\therefore \;a^2+b^2+2ga+2fb+c=0$-----(ii)

(i) cuts $x^2+y^2=4$ orthoganally

$\therefore 2(g \times 0 +f \times 0)=c-4$

$=>c=4$

from (ii) $a^2+b^2+2ga+2fb+4=0$

Locus of center $(-g,-f)$ is

$a^2+b^2-2ax+2by+4=0$

or $2ax+2by=a^2+b^2+4$

Hence B is the correct answer.

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