The parabolas $\mathrm{y}={\mathrm{x}}^2-9$ and $\mathrm{y}={\mathrm{\lambda x}}^2$ intersect at points A and B. If the length of AB is equal to 2a and if ${\mathrm{\lambda a}}^2+\mathrm{\mu }={\mathrm{a}}^2$ then the value of $\mathrm{\mu }$ is  
 

The parabolas are $\mathrm{y}={\mathrm{x}}^2-9$ and $\mathrm{y}={\mathrm{\lambda x}}^2$
$$\begin{gathered} {\mathrm{x}}^2-9={\mathrm{\lambda x}}^2\Rightarrow {\mathrm{x}}^2(1-\mathrm{\lambda })=9 \\ {\mathrm{x}}^2=\frac{9}{1-\mathrm{\lambda }}\Rightarrow \mathrm{x}=\frac{\pm 3}{\sqrt{1-\mathrm{\lambda }}} \end{gathered}$$
Now from the symmetry about y-axis
$$\begin{gathered} \mathrm{AB}=2\mathrm{a}=\frac{6}{\sqrt{1-\mathrm{\lambda }}}\Rightarrow \mathrm{a}=\frac{3}{\sqrt{1-\mathrm{\lambda }}} \\ {\mathrm{a}}^2(1-\mathrm{\lambda })=9\to {\mathrm{\lambda a}}^2+9={\mathrm{a}}^2 \\ \lambda {\mathrm{a}}^2+\mathrm{\mu }={\mathrm{a}}^2\Rightarrow \mathrm{\mu }=9 \end{gathered}$$