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 ${}^{\prime }{\alpha }^{\prime }$ ?

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

Answer 1

The correct option is B

Abscissae of foci

Given equation of hyperbola is $\frac{x^2}{co{s}^2\alpha }-\frac{y^2}{si{n}^2\alpha }=1$
Here, $a^2=co{s}^2\alpha$ and $b^2=si{n}^2\alpha$
[i.e. comparing with standard equation $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$]
We know that, foci $=(\pm ae,0)$
where, $ae=\sqrt{a^2+b^2}=\sqrt{co{s}^2\alpha +si{n}^2\alpha }=1$
$\Rightarrow Foci=(\pm 1,0)$
where, vertices are $(\pm cos\alpha ,0)$
Eccentricity, $ae=1ore=\frac{1}{cos\alpha }$
Hence, foci remains constant with change in $\alpha$.