Let a and b respectively be the semitransverse and semi-conjugate axes of a
hyperbola whose eccentricity satisfies the equation 9e2 - 18e + 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 :
Solutions · 1
Official solution
As S(5, 0) is the focus.
\[ \therefore \] ae = 5 . . . (1)
As 5x = 9
\[ \therefore \] x = \[{9 \over 5}\] is the directrix
As we know directrix = \[{a \over e}\]
\[ \therefore \] \[{a \over e} = {9 \over 5}\] . . . .(2)
Solving (1) and (2), we get
a = 3 and e = \[{5 \over 3}\]
As we know,
b2 = a2 (e2 - 1) = 9\[\left( {{{25} \over 9} - 1} \right)\] = 16
\[ \therefore \] a2 - b2 = 9 - 16 = - 7