Let a and b respectively be the semi-transverse and semi-conjugate axes of a hyperbola whose eccentricity satisfies the equation 9e² - 8e + 5 = 0. If S(5,0) is a focus and 5x = 9 is the corresponding directrix of this hyperbola, then a² - b² is equal to:

Length of conjugate axis = 2b
Length of transverse axis = 2a
ae = 5 − (1)
Substituting e from equation (1), a/5 = 9/5
a² = 9
9e² - 8e + 5 = 0
9e² - 15e + 7e + 5 = 0
3e(3e - 5) + 1(3e -5) = 0
(3e - 5)(3e + 1) = 0
e = 5/3 and e = -1/3
Since eccentricity is always positive, e = 5/3
We have ae = 5
a(5/3) = 5
a = 3
b² = a²(e² - 1) = 9((25/9) - 1) = 9(16/9) = 16
a² - b² = 9 - 16 = -7