Question

If `e_1` is the eccentricity of the ellipse `x^2/16+y^2/25=1 and e_2` is the eccentricity of the hyperbola passing through the foci of the ellipse and `e_1 e_2=1`, then equation of the hyperbola is
A. `(x^(2))/(9)-(y^(2))/(16)=1`
B. `(x^(2))/(16)-(y^(2))/(9)=-1`
C. `(x^(2))/(9)-(y^(2))/(25)=1`
D. none of these

Answer 1

Wehave, `e_(1)=sqrt(1-(16)/(25))=(3)/(5)`
`:.e_(1)e_(2)=1impliese_(2)=(5)/(3)`
The coordinates of foci of the ellipse are `(0,+-3)`.
Clearly, hyperbola in option `(b)` passes through `(0,+-3)` and has eccentricity `(5)/(3)`.