For some $\theta \in \left(0, \frac{\pi}{2}\right),$ if the eccentricity of the hyperbola $x^{2} - y^{2} \sec^{2} \theta = 10$ is $\sqrt{5}$ times the eccentricity of the ellipse $x^{2} \sec^{2} \theta + y^{2} = 5,$ then the length of the latus rectum of the ellipse is

The normal at $\left( 2, \frac{3}{2} \right)$ to the ellipse $\frac{x^2}{16} + \frac{y^2}{3} = 1$ touches a parabola,whose equation is

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 . . . . . . .

If $e$ and $e'$ are the eccentricities of the ellipse $5x^2 + 9y^2 = 45$ and the hyperbola $5x^2 - 4y^2 = 45$ respectively,then $ee' = $

The angle between the curves $x^2-y^2=4$ and $x^2+y^2=4\sqrt{2}$ is

The product of the slopes of the common tangents to the ellipse $\frac{x^2}{32} + \frac{y^2}{8} = 1$ and the parabola $y^2 = 8x$ is: