Question

For the hyperbola $\frac{x^2}{co{s}^2\alpha }-\frac{y^2}{si{n}^2\alpha }=1$, which of the following remains constant when α varies?

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

Answer 1

The correct option is D Abscissae of foci

$\frac{x^2}{co{s}^2\alpha }-\frac{y^2}{si{n}^2\alpha }=1Wehavesi{n}^2\alpha =co{s}^2\alpha (e^2-1)\Rightarrow ta{n}^2\alpha =c^2-1\Rightarrow e^2=se{c}^2\alpha \Rightarrow e=sec\alpha \to \left(1\right)$
Equation of directrix is $x=\frac{a}{e}\Rightarrow x=\frac{cos\alpha }{sec\alpha }\Rightarrow x=co{s}^2\alpha \to \left(2\right)$
Vertices = $(\pm cos\alpha ,0)\to \left(3\right)$
$Foci=(\pm ae,0)=(\pm cos\alpha ,sec\alpha ,0)=(\pm 1,0)\to \left(4\right)$
From (1), (2), (3) and (4)
(4) is required answer