Let the line $2\mathrm{y}=-\mathrm{x}+\mathrm{k}$ is tangent to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{11}=1\left(a^2>11\right)$which cuts its auxiliary circle at points $A$and$B$ such that $⌊\mathrm{AOB}={90}^{\circ }$ If "$e$" is eccentricity of ellipse, then the value of $8{\mathrm{e}}^2$ ___ (O being origin)