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 $y=2x+c$ 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)

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