The point $(2\mathrm{a}, \mathrm{a})$ lies inside the region bounded by the parabola ${\mathrm{x}}^2=4\mathrm{y}$ and its latus rectum. Then, 

$$\begin{gathered} \mathrm{S}={\mathrm{x}}^2=4\mathrm{y}\left(2\mathrm{a}, \mathrm{a}\right) \\ {\mathrm{S}}_{11}=4{\mathrm{a}}^2-4\mathrm{a} <0 \\ 4\mathrm{a}\left(\mathrm{a}-1\right)<0 \\ \Rightarrow \mathrm{a}(\mathrm{a}-1)<0\to \left(1\right) \end{gathered}$$

Also, the vertex A(0,0) and the point (2a,a) are on the same side of the line y=1 (the equation of the latus rectum)

So, $L_{11}.L_{22}>0$

$\Rightarrow$a−1<0⇒a<1   ....(2)

From (1) and (2), we have a(a−1)<0⇒0<a<1