The eccentricity of the hyperbola conjugate to ${x^2} - 3{y^2} = 2x + 8$ is

If $5x + 9 = 0$ is the directrix of the hyperbola $16x^2 -9y^2 = 144,$ then its corresponding focus is

If the line $y=m x+c$ is a common tangent to the hyperbola $\frac{x^{2}}{100}-\frac{y^{2}}{64}=1$ and the circle $x^{2}+y^{2}=36,$ then which one of the following is true?

Find the equation of the hyperbola with foci $(0,\,\pm 3)$ and vertices $(0,\,\pm \frac {\sqrt {11}}{2})$.

The eccentricity of the hyperbola whose length of the latus rectum is equal to $8$ and the length of its conjugate axis is equal to half of the distance between its foci is :

lf $e_1$ , $e_2$ and $e_3$ are eccentricities of the conics $y = {x^2} - x + 3,\,\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{3{a^4}}} = 1$ and ${a^2}{x^2} - 3{a^4}{y^2} = 1$ respectively, then which of the following is correct ? (where $a > 1)$