Since the mid-point of the line segment $MN$ lie on the given line,

 $\Rightarrow \frac{\lambda }{2}+\frac{{\lambda }^2+2}{2}={\lambda }^2\dots ..\left(\mathrm{i}\right)$

Also, the line segment $MN$ is perpendicular to the given line
$\Rightarrow \left(\frac{{\lambda }^2-2}{\lambda -0}\right)\times \left(-1\right)=-1\dots ..\left(\mathrm{i}\mathrm{i}\right)$

As $\left(\mathrm{i}\right)$ and $\left(\mathrm{i}\mathrm{i}\right)$ must be satisfied simultaneously,
$\therefore$ On solving, we get,

 $$\begin{gathered} \lambda =-\mathrm{1,2} \\ \Rightarrow {\lambda }_1^2+{\lambda }_2^2=1+4=5 \end{gathered}$$