Question

For the hyperbola ,$\frac{x^2}{{\cos }^{2}a}-\frac{y^2}{{\sin }^{2}a}=1$which of the following remains constant when $ɑ$ varies?

• A. Eccentricity
• B. Directrix
• C. Abscissae of vertices
• D. Abscissae of foci

Answer 1

The correct option is D

Abscissae of foci

Explanation for the correct options:

Finding the value.

In hyperbola:

$b^2=a^2(e^2–1)$

$\Rightarrow$ ${\sin }^{2}a=co{s}^{2}a(e^2-1)$

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

$\Rightarrow$ $e^2=se{c}^{2}a$

So, Vertices $\equiv (\pm \cos \alpha ,0)$

Coordinate of foci are $(\pm ae,0)\equiv (\pm 1,0)$

If $ɑ$ varies then the abscissa of foci remain constant.

Hence, Option(D) is correct.

Explanation for the incorrect options:

If $ɑ$ varies then Eccentricity, Directrix and Abscissae of vertices does not remain constant.

Hence, Option(A),(B) and (C) are incorrect.