Number of intersecting points of the conics $4x^{2} + 9y^{2} = 1$ and $4x^{2} + y^{2} = 4$ is

If the curves $\frac{x^2}{\alpha} + \frac{y^2}{4} = 1$ and $y^3 = 16x$ intersect at right angles,then a value of $\alpha$ is

An ellipse $E: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ passes through the vertices of the hyperbola $H: \frac{x^{2}}{49}-\frac{y^{2}}{64}=-1$. The major and minor axes of the ellipse $E$ coincide with the transverse and conjugate axes of the hyperbola $H$. Let the product of the eccentricities of $E$ and $H$ be $\frac{1}{2}$. If $l$ is the length of the latus rectum of the ellipse $E$,then the value of $113l$ is equal to $....$

If the extremities of the latus rectum having positive ordinate of the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ $(a > b)$ lie on the parabola $x^2 + 2ay - 4 = 0$,then the points $(a, b)$ lie on the curve:

The slope of a common tangent to the ellipse $\frac{x^2}{49}+\frac{y^2}{4}=1$ and the circle $x^2+y^2=16$ is

$AB$ is a double ordinate of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ such that $\Delta AOB$ (where $O$ is the origin) is an equilateral triangle. Then the eccentricity $e$ of the hyperbola satisfies: