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

$A$ tangent is drawn to the parabola $y^2 = 4x$ at the point $P(t^2, 2t)$,where the abscissa $t^2$ lies in the interval $[1, 4]$. The maximum possible area of the triangle formed by the tangent at $P$,the ordinate of the point $P$,and the $x$-axis is equal to

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 the same as the focus of $H$ with positive abscissa,and the directrix of the parabola passes through the other focus of $H$. If the length of the latus rectum of the parabola is $e$ times the length of the latus rectum of $H$,where $e$ is the eccentricity of $H$,then which of the following points lies on the parabola?

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 $B$. The orthocenter of $\Delta PAB$ is:

The angle between the curves $2x^2 + y^2 = 20$ and $4y^2 - x^2 = 8$ at a point where they intersect in the $4^{th}$ quadrant is

If the product of eccentricities of the ellipse $\frac{x^2}{16}+\frac{y^2}{b^2}=1$ and the hyperbola $\frac{x^2}{9}-\frac{y^2}{16}=-1$ is $1$,then $b^2=$