The value of $\lambda$ for which $(\lambda ,\lambda -2)$ lies inside of the ellipse $4x^2+9y^2=36$ and out side the parabola $y^2=x$ satisfy 

$\begin{array}{l}4{\lambda }^2+9\left({\lambda }^2+4-4\lambda \right)-36>0\\ 13{\lambda }^2-36\lambda <0\\ \lambda (13\lambda -36)>0\\ \lambda \in \left(0,\frac{36}{13}\right)---\left(1\right)\end{array}$
And ${\lambda }^2+4-4\lambda -\lambda >0$
$\begin{array}{l}{\lambda }^2-5\lambda +4>0\\ (\lambda -1)(\lambda -4)>0\\ \lambda <1\left(\text{ or }\right)\lambda >4--\left(2\right)\end{array}$
From (1) and (2) $0<\lambda <1$