The line $3x - 4y = 5$ is a tangent to the hyperbola $x^2 - 4y^2 = 5$. The point of contact is

The length of the latus rectum of the hyperbola $\frac{x^2}{\cos^2 \alpha} - \frac{y^2}{\sin^2 \alpha} = 4$ is (where $\alpha \neq \frac{n\pi}{2}, n \in I$).

The locus of the middle points of the chords of the hyperbola $3x^2 - 2y^2 + 4x - 6y = 0$ parallel to $y = 2x$ is

Statement $I$: The eccentricity of the hyperbola $9x^2-16y^2-72x+96y-144=0$ is $5/4$.
Statement $II$: The eccentricity of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ is $\sqrt{1+\frac{b^2}{a^2}}$.

For what value of $m$ is the line $y = mx + 6$ a tangent to the hyperbola $\frac{x^2}{100} - \frac{y^2}{49} = 1$?

The number of integral points $(x, y)$ interior to the circle $x^2 + y^2 = 10$ from which exactly one real tangent can be drawn to the curve $\sqrt{(x + 5\sqrt{2})^2 + y^2} - \sqrt{(x - 5\sqrt{2})^2 + y^2} = 10$ is (where an integral point $(x, y)$ means $x, y \in \mathbb{Z}$):