The angle between the tangents drawn from $(1, 2\sqrt{2})$ to the hyperbola $16x^{2} - 25y^{2} = 400$ is.....

The equation $3x^2 + 7xy + 2y^2 + 5x + 5y + 2 = 0$ represents:

If $y=mx+4$ ($n>0$ is a typo,likely $m>0$ or irrelevant) is a tangent to the hyperbola $\frac{x^2}{25}-\frac{y^2}{9}=1$,then the point of contact of this tangent is

Let $m_1$ and $m_2$ be the slopes of the tangents drawn from the point $P(4, 1)$ to the hyperbola $H: \frac{y^2}{25} - \frac{x^2}{16} = 1$. If $Q$ is the point from which the tangents drawn to $H$ have slopes $|m_1|$ and $|m_2|$ and they make positive intercepts $\alpha$ and $\beta$ on the $x$-axis,then $\frac{(PQ)^2}{\alpha \beta}$ is equal to $............$.

The distance between the foci of the hyperbola $x^2 - 3y^2 - 4x - 6y - 11 = 0$ is

The distance between the tangents of the hyperbola $2x^2 - 3y^2 = 6$ which are perpendicular to the line $x - 2y + 5 = 0$ is