Question

Let the eccentricity of the hyperbola $H : \frac{ x ^{2}}{ a ^{2}}-\frac{ y ^{2}}{ b ^{2}}=1$ be $\sqrt{\frac{5}{2}}$ and length of its latus rectum be $6 \sqrt{2}$, If $y=2 x+c$ is a tangent to the hyperbola $H$, then the value of $c ^{2}$ is equal to

• A. 18
• B. 20
• C. 24
• D. 32

Answer 1

Answer: (B)

$y = mx \pm \sqrt{ a ^{2} m ^{2}- b ^{2}}$
$m =2, c ^{2}= a ^{2} m ^{2}- b ^{2}$
$c ^{2}=4 a ^{2}- b ^{2}$
$e ^{2}=1+\frac{ b ^{2}}{ a ^{2}}$
$\frac{5}{2}=1+\frac{b^{2}}{a^{2}}$
$\frac{3}{2}=\frac{b^{2}}{a^{2}}$
$\Rightarrow b^{2}=\frac{3 a^{2}}{2}$
$\frac{2 b^{2}}{a}=6 \sqrt{2}$
$\frac{2}{a} \times \frac{3 a^{2}}{2}=6 \sqrt{2}$
$3 a=6 \sqrt{2}$
$a =2 \sqrt{2}$
$b ^{2}=\frac{3}{2} \times 8=12$
$b =2 \sqrt{3}$
$\therefore c ^{2}=4 \times 8-12$
$c ^{2}=20$