Explanation

(a)
Let the equation of circle be
$x^{2} + y^{2} + \ 2gx + 2fy + c = 0$ …(i)
Given equation of circles are
$x^{2} + y^{2} - 2x + 3y - 7 = 0$ …(ii)
$x^{2} + y^{2} + 5x - 5y + 9 = 0$ …(iii)
and $x^{2} + y^{2} + 7x - 9y + 29 = 0$ …(iv)
Since, the circle (i) cut all three circles orthogonally,
$\therefore\ \ \ 2g( - 1) + 2f(3/2) = c - 7\ \ \Rightarrow \ - 2g + 3f - c = - 7$ …(v)
$2g(5/2) + 2f( - 5/2) = c + 29\ \ \ \Rightarrow \ \ 5g - 5f - c = 9$ …(vi)
$2g\left( \frac{7}{2} \right) + 2f\left( - \frac{9}{2} \right) = c + 29\ \ \Rightarrow 7g - 9f - c = 29$ …(vii)
On solving Eqs. (v), (vi) and (vii), we get
$g = - 8,\ \ \ f = - 9$ and $c = - 4$
On putting the values of $g,\ f$ and $c$ in Eq. (i), we get
$x^{2} + y^{2} - 16x - 18y - 4 = 0$