Explanation

(a)
Since, the focus and vertex of the parabola are on $y$-axis, therefore its axis of the parabola is $y -$ axis
Let the equation of the directrix be $y = k$ the directrix meets the axis of the parabola at $(0,\ k)$. But vertex is the mid point of the line segment joining the focus to the point where directrix meets axis of the parabola
$k + \frac{3}{2} = 6\ \Rightarrow \ k = 9$
Thus, the equation of directrix is $y = 9$
Equation of parabola is
$(x - 0)^{2} + (y - 3)^{2} = (y - 9)^{2}$
$\Rightarrow \ x^{2} + 12y - 72 = 0$