Explanation

(a)
Required equation of circle is
$x^{2} + y^{2} - 6x - 8y + \lambda(x + y - 1) = 0$
$\Rightarrow \ x^{2} + y^{2} - (6 - \lambda)x - (8 - \lambda)y - \lambda) = 0$
Whose centre is $\left( 3 - \frac{\lambda}{2},\ 4 - \frac{\lambda}{2} \right)$
Which lies on the line $x + y - 1 = 0$
$\Rightarrow \ 3 - \frac{\lambda}{2} + 4 - \frac{\lambda}{2} - 1 = 0$
$\Rightarrow \ \lambda = 6$
Hence, required equation is
$x^{2} + y^{2} - 6x - 8y + 6x + 6y - 6 = 0$
$\Rightarrow \ x^{2} + y^{2} - 2y - 6 = 0$