Question

Let a and b respectively be the semitransverse 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 :

• A. 7
• B. - 7
• C. 5
• D. -5

Answer 1

Answer: (B)

ae $= 5$
$x=\frac{9}{5} \Rightarrow \frac{a}{e}=\frac{9}{5}$
$\Rightarrow a^{2}=9$
$\left(1\right) \Rightarrow e=\frac{5}{3}$
$b^{2}=a^{2}\left(e^{2}-1\right) \Rightarrow b^{2}=16$
$a^{2}-b^{2}=9-16=-7$