Let \( A\!\left(\dfrac{3}{\sqrt{a}}, \sqrt{a}\right) \), \( a>0 \), be a fixed point in the \( xy \)-plane. The image of \( A \) in \( y \)-axis be \( B \) and the image of \( B \) in \( x \)-axis be \( C \). If \( D(3\cos \theta, a \sin \theta) \) is a point in the fourth quadrant such that the maximum area of \( \triangle ACD \) is \( 12 \) square units, then \( a \) is equal to _________.
