Let a line L_1 be tangent to the hyperbola x^2/16-y^2/4=1 and let L_2 be the line passing through the origin and perpendicular to L_1. If the locus…

Let a line $\displaystyle L_1$ be tangent to the hyperbola $\displaystyle \frac{x^2}{16}-\frac{y^2}{4}=1$ and let $\displaystyle L_2$ be the line passing through the origin and perpendicular to $\displaystyle L_1$. If the locus of the point of intersection of $\displaystyle L_1$ and $\displaystyle L_2$ is $\displaystyle \left(x^2+y^2\right)^2=\alpha x^2+\beta y^2$, then $\displaystyle \alpha+\beta$ is equal to \_\_\_\_.