In the figure, Statement $I$: when $\alpha > \beta \ge 0$, the section is a hyperbola. Statement $II$: when $\beta > 90^\circ$, the section is an ellipse.

The points of intersection of the curves whose parametric equations are $x = t^2 + 1, y = 2t$ and $x = 2s, y = \frac{2}{s}$ is given by

If $S \equiv \frac{x^2}{k-7}+\frac{y^2}{11-k}-1=0, k \in R-\{7,11\}$,then which one of the following statements is incorrect?

Tangents are drawn from the point $P\ (3, 4)$ to the ellipse $\frac{x^2}{9} + \frac{y^2}{4} = 1$,touching the ellipse at points $A$ and $B$. Find the coordinates of $A$ and $B$.

Suppose $A, B, C$ and $D$ are the four intersection points of the curves $\frac{x^2}{18}+\frac{y^2}{8}=1$ and $x^2-y^2=5$ in $I, II, III$ and $IV$ quadrants respectively. If $\theta_1, \theta_2, \theta_3$ and $\theta_4$ respectively are the angles between the curves at $A, B, C$ and $D$, then

The eccentricity of an ellipse $E$ with centre at the origin $O$ is $\frac{\sqrt{3}}{2}$ and its directrices are $x = \pm \frac{4\sqrt{6}}{3}$. Let $H : \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ be a hyperbola whose eccentricity is equal to the length of semi-major axis of $E$, and whose length of latus rectum is equal to the length of minor axis of $E$. Then the distance between the foci of $H$ is :