Hyperbola

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 passes through the foci of the ellipse ${{{x^2}} \over {25}} + {{{y^2}} \over {16}} = 1$ and its transverse and conjugate axes coincide with major and minor axes of the ellipse, respectively. If the product of their eccentricities is one, then the equation of the hyperbola is :

Let a > 0, b > 0. Let e and l respectively be the eccentricity and length of the latus rectum of the hyperbola ${{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1$. Let e' and l' respectively be the eccentricity and length of the latus rectum of its conjugate hyperbola. If ${e^2} = {{11} \over {14}}l$ and ${\left( {e'} \right)^2} = {{11} \over 8}l'$, then the value of $77a + 44b$ is equal to :

Let the foci of the ellipse $\frac{x^{2}}{16}+\frac{y^{2}}{7}=1$ and the hyperbola $\frac{x^{2}}{144}-\frac{y^{2}}{\alpha}=\frac{1}{25}$ coincide. Then the length of the latus rectum of the hyperbola is :

If the line $x-1=0$ is a directrix of the hyperbola $k x^{2}-y^{2}=6$, then the hyperbola passes through the point :

Let the hyperbola $H: \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$ pass through the point $(2 \sqrt{2},-2 \sqrt{2})$. A parabola is drawn whose focus is same as the focus of $\mathrm{H}$ with positive abscissa and the directrix of the parabola passes through the other focus of $\mathrm{H}$. If the length of the latus rectum of the parabola is e times the length of the latus rectum of $\mathrm{H}$, where e is the eccentricity of H, then which of the following points lies on the parabola?

Let $\mathrm{H}$ be the hyperbola, whose foci are $(1 \pm \sqrt{2}, 0)$ and eccentricity is $\sqrt{2}$. Then the length of its latus rectum is :

For $0<\theta<\pi / 2$, if the eccentricity of the hyperbola $x^2-y^2 \operatorname{cosec}^2 \theta=5$ is $\sqrt{7}$ times eccentricity of the ellipse $x^2 \operatorname{cosec}^2 \theta+y^2=5$, then the value of $\theta$ 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 :

If the foci of a hyperbola are same as that of the ellipse $\frac{x^2}{9}+\frac{y^2}{25}=1$ and the eccentricity of the hyperbola is $\frac{15}{8}$ times the eccentricity of the ellipse, then the smaller focal distance of the point $\left(\sqrt{2}, \frac{14}{3} \sqrt{\frac{2}{5}}\right)$ on the hyperbola, is equal to