Question

Which one of the following is independent of $\alpha$ in the hyperbola $(0<\alpha <\pi /2)$ $\frac{x^2}{{\cos }^2\alpha }-\frac{y^2}{{\sin }^2\alpha }=1$

• A. Eccentricity
• B. Abscissa of foci
• C. Directrix
• D. Vertex

Answer 1

The correct option is B Abscissa of foci

$\frac{x^2}{{\cos }^2\alpha }-\frac{y^2}{{\sin }^2\alpha }=1e^2=1+\frac{a^2}{b^2}{e}^2=1+\frac{{\sin }^2\alpha }{{\cos }^2\alpha }{e}^2=\frac{{\cos }^2\alpha +{\sin }^2\alpha }{{\cos }^2\alpha }=\frac{1}{{\cos }^2\alpha }e=\frac{1}{\cos \alpha }$

Coordinates of foci are $(\pm ae,0)$

So, the coordinates of the foci are

$(\cos \alpha \times \frac{1}{\cos \alpha },0)\Rightarrow (\pm 1,0)$

Hence, the abscissa of foci is independent of $\alpha$

So, option C is correct.