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

Let the eccentricity of the hyperbola $${{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1$$ be $${5 \over 4}$$. If the equation of the normal at the point $$\left( {{8 \over {\sqrt {5} }},{{12} \over {5}}} \right)$$ on the hyperbola is $$8\sqrt 5 x + \beta y = \lambda $$, then $$\lambda$$ $$-$$ $$\beta$$ is equal to ___________.

Let the hyperbola $$H:{{{x^2}} \over {{a^2}}} - {y^2} = 1$$ and the ellipse $$E:3{x^2} + 4{y^2} = 12$$ be such that the length of latus rectum of H is equal to the length of latus rectum of E. If $${e_H}$$ and $${e_E}$$ are the eccentricities of H and E respectively, then the value of $$12\left( {e_H^2 + e_E^2} \right)$$ is equal to ___________.

^{2}$$-$$ y

^{2}= 2. If ($$\alpha$$, $$\beta$$) is the point of the intersection of the normals to the hyperbola at A and B, then (2$$\beta$$)

^{2}is equal to ____________.