Step-by-step Solution:

Equation of the hyperbola: \[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \] From the equation, \[ b^2 = a^2 (e^2 - 1) \] \[ \Rightarrow e^2 = 1 + \frac{b^2}{a^2} = \frac{a^2 + b^2}{a^2} \] Equation of the conjugate hyperbola: \[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = -1 \] Rearranging, \[ \frac{y^2}{b^2} - \frac{x^2}{a^2} = 1 \] Similarly, \[ e'^2 = 1 + \frac{a^2}{b^2} = \frac{a^2 + b^2}{b^2} \] Now, \[ \frac{1}{e^2} + \frac{1}{e'^2} = \frac{b^2}{a^2 + b^2} + \frac{a^2}{a^2 + b^2} = 1 \]