lf the eccentricity of the hyperbola `x^2 - y^2 sec^2 alpha=5` is `sqrt3` times the eccentricity of the ellipse `x^2 (sec^2alpha )+y^2=25,` then a value of `alpha` is : (a) `pi/6` (b) `pi/4` (c) `pi/3` (d) `pi/2`

Options

A. `pi//6`
B. `pi//4`
C. `pi//3`
D. `pi//2`

Answer

B

Explanation

For the hyperbola
`(x^(2))/(5)-(y^(2))/(5 cos^(2)alpha)=1`
we have
`e_(1)^(2)=1+(b^(2))/(a^(2))=1+(5 cos^(2)alpha)/(5)=1+cos^(2)alpha`
For the ellipse
`(x^(2))/(25 cos^(2)alpha)+(y^(2))/(25)=1`
we have
`e_(2)^(2)=1-(25 cos^(2)alpha)/(25)=sin^(2)alpha`
Given that
`e_(1)=sqrt3e_(2)`
`therefore" "e_(1)^(2)=3e_(2)^(2)`
`"or "1+cos^(2)alpha=3 sin^(2)alpha`
`"or "2=4 sin^(2)alpha`
`"or "sin alpha=(1)/(sqrt2)`