Let the eccentricity of the hyperbola `(x^(2))/(b^(2))-(y^(2))/(b^(2))=1` be reciprocal to that of the ellipse `x^(2)+4y^(2)=4`. If the hyperbola passes through a focus of the ellipse, then

Options

A. the equation of the hyperbola is `(x^(2))/(3)-(y^(2))/(2)=1`
B. a focus of the hyperbola is (2, 0)
C. the eccentricity of the hyperbola is `sqrt((5)/(3))`
D. the equation of the hyperbola is `x^(2)-3y^(2)=3`

Explanation

Here, equation of ellipse is `(x^(2))/(4)+(y^(2))/(1)=1`
`implies e^(2) =1-(b^(2))/(a^(2))=1-(1)/(4)=(3)/(4)`
` therefore e=(sqrt(3))/(2) " and focus " (pm a e, 0) rArr (pm sqrt(3),0)`
For hyperbola `(x^(2))/(a^(2))-(y^(2))/(b^(2))=1, e_(1)^(2)=1+(b^(2))/(a^(2))`
where, `e_(1)^(2)=(1)/(e^(2))=(4)/(3) rArr 1+(b^(2))/(a^(2))=(4)/(3)`
` therefore (b^(2))/(a^(2))=(1)/(3) " ...(i)" `
and hyperbola passes through `(pm sqrt(3),0)`
`rArr (3)/(a^(2))=1 rArr a^(2) =3 " ...(ii) " `
From Eqs. (i) and (ii), `b^(2)=1 " ...(iii)" `
` therefore " Equation of hyperbola is " (x^(2))/(3)-(y^(2))/(1)=1`
Focus is `(pm a e_(1),0) rArr (pm sqrt(3) *(2)/(sqrt(3)),0) rArr (pm 2, 0)`
Hence, (b) and (d) are correct answers.