Explanation

(b)
The required equation of circle is
$\left( x^{2} + y^{2} - 6 \right) + \lambda\left( x^{2} + y^{2} - 6y + 8 \right) = 0$ …(i)
It passes through (1, 1)
$\therefore\ \ \ (1 + 1 - 6) + \lambda(1 + 1 - 6 + 8) = 0$
$\Rightarrow \ - 4 + 4\lambda = 0$
$\Rightarrow \ \lambda = 1$
$\therefore$ required equation of circle is
$x^{2} + y^{2} - 6 + x^{2} + y^{2} - 6y + 8 = 0$
$\Rightarrow \ \ 2x^{2} + 2y^{2} - 6y + 2 = 0$
$\Rightarrow \ \ x^{2} + y^{2} - 3y + 1$