Let a and b respectively be the semi-transverse and semi-conjugate axes of a standard hyperbola whose eccentricity satisfies the equation 9 e 2 - 18 e + 5 = 0 . If S 5 , 0 is a focus and 5 x = 9 is the corresponding directrix of this hyperbola, then a 2 - b 2 is equal to

Step-by-step solution

Given that focus of hyperbola is S 5 , 0 ⇒ a e = 5 ...... i Therefore, Directrix x = a e ⇒   a e = 9 5 ...... ii From i and ii , we get ⇒     a 2 = 9 ⇒ a = 3 and e = 5 3 (It satisfies 9 e 2 - 18 e + 5 = 0 ) We also know that for an ellipse, b 2 = a 2 e 2 - 1 ⇒ b 2 = 16 ⇒ a 2 - b 2 = 9 - 16 =   - 7