The line passing through A,B where A and B are ends of major and minor axes of an ellipse$4{\mathrm{x}}^2+{\mathrm{y}}^2=4$   meets auxiliary circle at the point  M then area of $\mathrm{\Delta OAM},$ O is origin is$\frac{4\mathrm{k}}{5}$  . Then  k is

For given ellipse  end points of major and minor axes are

A(0,2),B(1,0)

Equation of AB in $\frac{\mathrm{x}}{1}+\frac{\mathrm{y}}{2}=1\Rightarrow \mathrm{y}=2-2\mathrm{x}$
Equation of auxiliary circle is 
$\begin{array}{l}{\mathrm{x}}^2+{\mathrm{y}}^2=4\Rightarrow {\mathrm{x}}^2+(2-2\mathrm{x}{)}^2=4\\ \Rightarrow 5{\mathrm{x}}^2-8\mathrm{x}=0\Rightarrow \mathrm{x}=\frac{8}{5}\mathrm{y}=\frac{-6}{5}\end{array}$
then, 
$\begin{array}{l}\mathrm{M}=\left(\frac{8}{5},\frac{-6}{5}\right)\\ \mathrm{A}=(0,2),O=(0,0)\end{array}$
$\therefore \text{ Area of }\mathrm{△}\mathrm{OAM}=\frac{8}{5}$