For the hyperbola `(x^2)/(cos^2alpha)-(y^2)/(sin^2alpha)=1` , which of the following remains constant when `alpha` varies? (1) eccentricity (2) directrix (3) abscissae of vertices (4) abscissae of foci

Options

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

Answer

D

Explanation

Here, `a^(2)=cos^(2)alpha and b^(2)=sin^(2)alpha`
` Now, `e=sqrt(1+b^(2)/(a^(2)))rArre=sqrt(1+(sin^(2)alpha)/(cos^(2)alpha)`
` rArr e=sqrt(1+tan^(2)alpha)` rArre=secalpha` coordinates of foci are `(+-ae,0),i.e.(+-1,0).` hence, abscissaen of foci remains constant when `alpha` varies.