The equation $x^2 - 4y^2 - 2x + 16y - 40 = 0$ represents:

$A$ tangent to the curve $9 b^2 x^2 - 4 a^2 y^2 = 36 a^2 b^2$ makes intercepts of unit length on each of the coordinate axes. Then the point $(a, b)$ lies on:

The equation of a tangent to the hyperbola $5x^{2}-y^{2}=5$ which passes through the external point $(2, 8)$ is:

If $e_{1}$ and $e_{2}$ are the eccentricities of a hyperbola $3x^{2} - 3y^{2} = 25$ and its conjugate,then

The equation of the hyperbola with asymptotes $3x - 4y + 7 = 0$ and $4x + 3y + 1 = 0$ and which passes through the origin is

The latus-rectum of the hyperbola $16x^2 - 9y^2 = 144$ is