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

The square of the slope of a common tangent drawn to the circle $4x^2 + 4y^2 = 25$ and the ellipse $4x^2 + 9y^2 = 36$ is

At the point of intersection of the rectangular hyperbola $xy = c^2$ and the parabola $y^2 = 4ax$,tangents to the rectangular hyperbola and the parabola make an angle $\theta$ and $\phi$ respectively with the $X$-axis. Then:

For some $\theta \in (0, \frac{\pi}{2})$, let the eccentricity and the length of the latus rectum of the hyperbola $x^{2} - y^{2} \sec^{2} \theta = 8$ be $e_{1}$ and $l_{1}$, respectively, and let the eccentricity and the length of the latus rectum of the ellipse $x^{2} \sec^{2} \theta + y^{2} = 6$ be $e_{2}$ and $l_{2}$, respectively. If $e_{1}^{2} = e_{2}^{2}(\sec^{2} \theta + 1)$, then $(\frac{l_{1}l_{2}}{e_{1}e_{2}}) \tan^{2} \theta$ is equal to . . . . . . .

Let $P$ be the point of intersection of the common tangents to the parabola $y^2 = 12x$ and the hyperbola $8x^2 - y^2 = 8$. If $S$ and $S'$ denote the foci of the hyperbola where $S$ lies on the positive $x$-axis,then $P$ divides $SS'$ in the ratio:

Let $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, a > b$ be an ellipse,whose eccentricity is $\frac{1}{\sqrt{2}}$ and the length of the latus rectum is $\sqrt{14}$. Then the square of the eccentricity of $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ is: