If the points of the intersection of the two ellipses, \[ \dfrac {x^2} {a^2} + \dfrac{y^2} {b^2} = 1 , \quad \text{ and } \quad \dfrac {x^2} {\alpha^2} + \dfrac{y^2} {\beta^2} = 1 \]

are at the extremities of the conjugate diamaters of the former, find the value of \( \dfrac{a^2}{\alpha^2} + \dfrac{b^2}{\beta^2} \).

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