Let a line \( L_1 \) be tangent to the hyperbola \( \dfrac{x^2}{16} - \dfrac{y^2}{4} = 1 \) 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 _______.
