Question

If eccentricity of the hyperbola

$\frac{x^2}{{\cos }^2\theta }-\frac{y^2}{{\sin }^2\theta }=1$

is more than $2$ when $\in (0,\pi /2)$ then values of length of latus rectum lies in the interval

• A. $(3,\infty )$
• B. $\left(1,\frac{3}{2}\right)$
• C. $\left(2,3\right)$
• D. $(-3,-2)$

Answer 1

The correct option is A

$(3,\infty )$

Finding the length of the latus rectum:

For hyperbola,

$\begin{array}{rcl}{e}^2& =& 1+\frac{b^2}{a^2}\\ & =& 1+{\tan }^2\theta \\ & =& se{c}^2\theta \end{array}$

$e>2$

$\Rightarrow$$sec\theta >2$

$\in (\pi /3,\pi /2)$

$\therefore$The length of the latus rectum is

$\begin{array}{rcl}2b^2/a& =& 2\tan \theta \sin \theta \\ & =& 2(>\sqrt{3})(>\sqrt{3}/2)\\ & >& \mathbf{3}\end{array}$

Hence, the correct answer is option (A)