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

Options

A. `-7`
B. `-5`
C. `5`
D. `7`

Explanation

We have, `9e^(2)-18e+5=0`
`implies(3e-1)(3e-5)=0impliese=(5)/(3)`[`:'e gt 1 :. 3e-1 ne 0`]
The coordinates of a focus and the equation of the cooresponding directrix are `(5,0)` and `x=(9)/(5)` respectively.
`:.ae=5` and `(a)/(e)=(9)/(5)`
`impliesaexx(a)/(e)=5xx(9)/(5)impliesa^(2)=9impliesa=3`
`:.b^(2)=a^(2)(e^(2)-1)impliesb^(2)=9((25)/(9)-1)=16`
Hence, `a^(2)-b^(2)=9-16=-7`