Let e₁ and e₂ be the eccentricities of the ellipse,
$${{{x^2}} \over {25}} + {{{y^2}} \over {{b^2}}} = 1$$(b < 5) and the hyperbola,
$${{{x^2}} \over {16}} - {{{y^2}} \over {{b^2}}} = 1$$ respectively satisfying e₁e₂ = 1. If $$\alpha $$
and $$\beta $$ are the distances between the foci of the
ellipse and the foci of the hyperbola
respectively, then the ordered pair ($$\alpha $$, $$\beta $$) is equal to :

A hyperbola having the transverse axis of length $$\sqrt 2 $$ has the same foci as that of the ellipse 3x² + 4y² = 12, then this hyperbola does not pass through which of the following points?

For some $$\theta \in \left( {0,{\pi \over 2}} \right)$$, if the eccentricity of the
hyperbola, x²–y²sec²$$\theta $$ = 10 is $$\sqrt 5 $$ times the
eccentricity of the ellipse, x²sec²$$\theta $$ + y² = 5, then the length of the latus rectum of the ellipse, is :

A line parallel to the straight line 2x – y = 0 is tangent to the hyperbola
$${{{x^2}} \over 4} - {{{y^2}} \over 2} = 1$$ at the point $$\left( {{x_1},{y_1}} \right)$$. Then $$x_1^2 + 5y_1^2$$ is equal to :