Explanation

(b)
Given equation of circle is
$x^{2} + y^{2} + 2gx + 2fy + k = 0$ …(i)
And equation of hyperbola is $xy = c^{2}$ …(ii)
From Eqs.(i) and (ii), we get
$x^{2} + \left( \frac{c^{2}}{x} \right) + 2gx + 2f\left( \frac{c^{2}}{x} \right) + k = 0$
$\Rightarrow \ \ x^{4} + 2gx^{3} + kx^{2} + 2fc^{2}x + c^{4} = 0$
$\therefore\ \$Sum of roots $= x_{1} + x_{2} + x_{3} + x_{4} = - \frac{2g}{1} = - 2g$