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For the hyperbola x^2/ cos^2 alpha - y^2...

For the hyperbola `x^2/ cos^2 alpha - y^2 /sin^2 alpha = 1 which of the following remains constant when alpha varies?

A

Abscissae of vertices

B

Abscissae of foci

C

Eccentricity

D

Directrix

Text Solution

Verified by Experts

Given equation of hyperbola is `(x^(2))/(cos^(2)alpha)-(y^(2))/(sin^(2)alpha)=1 `
Here, `a^(2)=cos^(2)alpha and b^(2)=sin^(2)alpha`
[i.e. comparing with standard equation `(x^(2))/(a^(2))-(y^(2))/(b^(2))=1`]
We know that, foci `=(pm ae, 0)`
where, `ae=sqrt(a^(2)+b^(2))=sqrt(cos^(2)alpha+sin^(2)alpha)=1`
`rArr "Foci"=(pm 1, 0)`
where, vertices are `(pm cos alpha,0).`
Eccentricity, `ae=1 or e=(1)/(cos alpha)`
Hence, foci remains constant with change in `alpha`.
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