Given that the equation of ellipse is

$x^2+\frac{y^2}{4}=1$, where $a^2=1, b^2=4$

So, equation of normal in parametric form can be written as

$\frac{x}{\cos \theta } -\frac{2y}{\sin \theta } =1-4 \dots \left(\mathrm{i}\right)$

Comparing with $2x-\frac{8}{3}\lambda y=-3$

$\frac{1}{\cos \theta }=2\Rightarrow \cos \theta =\frac{1}{2}\Rightarrow \sin \theta =\pm \frac{\sqrt{3}}{2}$

and  

$-\frac{2}{\sin {\displaystyle \theta }}=-\frac{8}{3}\lambda \Rightarrow \lambda =\frac{3}{4\sin \theta }$

So, $\lambda = \pm \frac{\sqrt{3}}{2}$