Question

If $\frac{x^2}{{\cos }^2\alpha }-\frac{y^2}{{\sin }^2\alpha }=1$ represents family of hyperbolas where ${}^{\prime }{\alpha }^{\prime }$ varies then -

• A. Distance between the foci is constant.
• B. Distance between the two directrieces is constant.
• C. Distance between the vertices is constant.
• D. Distances between focus and the corresponding directrix is constant.

Answer 1

Answer: (A)

Distance between the foci is constant.

$\Rightarrow$ $\frac{x^2}{{\cos }^2\alpha }-\frac{y^2}{{\sin }^2\alpha }=1$ [ Given ]

We get,

$a^2={\cos }^2\alpha$ and $b^2={\sin }^2\alpha$

Co-ordinates of foci $=(\pm ae,0)$

$\Rightarrow$ $e^2=1+\frac{b^2}{a^2}$

$=1+\frac{{\sin }^2\alpha }{{\cos }^2\alpha }$

$=\frac{{\cos }^2\alpha +{\sin }^2\alpha }{{\cos }^2\alpha }$

$=\frac{1}{{\cos }^2\alpha }$

$={\sec }^2\alpha$

$\Rightarrow$ $e^2={\sec }^2\alpha$

$\Rightarrow$ $e=\sec \alpha$

Co-ordinates of foci $=(\pm \cos \alpha \sec \alpha ,0)=(\pm \cos \alpha \times \frac{1}{\cos \alpha },0)$

$=(\pm 1,0)$

$\therefore$ We can see foci is independent on $\alpha$

$\therefore$ When $\alpha$ varies then distance between the foci is constant.