Question

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:

• A. $7$
• B. $5$
• C. $-7$
• D. $-5$

Answer 1

Answer: (C)

$-7$

Length of conjugate axis $=2b$
Length of transverse axis $=2a$
$ae=5$ -- (1)
Subsituting $e$ from equation (1),
$\frac{a^2}{5}=\frac{9}{5}$
$a^2=9$
$9e^2-18e+5=0$
$9e^2-15e-3e+5=0$
$e=\frac{1}{3}$ and $e=\frac{5}{3}$
$e=\frac{1}{3}$ (not possible)
$e=\frac{5}{3}$
Using $e=\sqrt{1+\frac{b^2}{a^2}},$
we get $b^2=16$
So, $a^2-b^2=-7$