The value of $m$ for which $y = mx + 6$ is a tangent to the hyperbola $\frac{x^2}{100} - \frac{y^2}{49} = 1$ is:

If the foci of a hyperbola are the same as that of the ellipse $\frac{x^2}{9}+\frac{y^2}{25}=1$ and the eccentricity of the hyperbola is $\frac{15}{8}$ times the eccentricity of the ellipse,then the smaller focal distance of the point $\left(\sqrt{2}, \frac{14}{3} \sqrt{\frac{2}{5}}\right)$ on the hyperbola is equal to

The equation of the hyperbola referred to its axes as axes of coordinate and whose distance between the foci is $16$ and eccentricity is $\sqrt{2}$ is:

The eccentricity of the conjugate hyperbola of the hyperbola ${x^2} - 3{y^2} = 1$ is

The equation of a tangent to the hyperbola $16x^2 - 25y^2 - 96x + 100y - 356 = 0$ which makes an angle $45^{\circ}$ with its transverse axis is

The equation of the tangent at the point $(a \sec \theta, b \tan \theta)$ of the conic $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ is: