Question

For hyperbola $\frac{x^2}{co{s}^2\alpha }-\frac{y^2}{si{n}^2\alpha }=1$, which of the following remains constant with change in '$\alpha$'

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

Answer 1

The correct option is B Abscissae of foci

Hyperbola is $\frac{x^2}{co{s}^2\alpha }-\frac{y^2}{si{n}^2\alpha }=1$,
Coordinates of vertices are $(\pm cos\alpha ,0)$, eccentricity of the hyperbola is $e=\sqrt{1+\frac{si{n}^2\alpha }{co{s}^2\alpha }}=\left|sec\alpha \right|$
$\therefore$ Coordinates of foci are thus $(\pm 1,0)$,which are independent of $\alpha$. Directrix is $x=\pm co{s}^2\alpha$