Explanation

(a)
Normal at a point $(m^{2},\ - 2m)$ on the parabola $y^{2} = 4x$ is given by $y = mx - 2m - m^{3}.$ If this is normal to the circle also, then it will passes through centre $( - 3,\ 6)$ of the circle
$\therefore 6 = - 3m - 2m - m^{3} \Rightarrow m = - 1$
Since, shortest distance between parabola and circle will occurs along common normal
$\therefore$ Shortest distance = distance between $(m^{2},\ - 2m)$
And centre $( - 3,\ 6) -$ radius of circle $= 4\sqrt{2} - 5$