Find the equations of the transverse axis and conjugate axis of the hyperbola $16x^2 - y^2 + 64x + 4y + 44 = 0$.

Consider a hyperbola $H : x^{2}-2y^{2}=4$. Let the tangent at a point $P(4, \sqrt{6})$ meet the $x$-axis at $Q$ and the latus rectum at $R(x_{1}, y_{1})$,where $x_{1}>0$. If $F$ is a focus of $H$ which is nearer to the point $P$,then the area of $\Delta QFR$ is equal to ....... .

Let $P(6,3)$ be a point on the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$. If the normal at the point $P$ intersects the $x$-axis at $(9,0)$,then the eccentricity of the hyperbola is

The equation of the hyperbola whose directrix is $x + 2y = 1$,focus $(2, 1)$ and eccentricity $e = 2$ is:

If in a hyperbola,the distance between the foci is $10$ and the transverse axis has length $8$,then the length of its latus rectum is

The descending order of magnitude of the eccentricities of the following hyperbolas is:
$A$. $A$ hyperbola whose distance between foci is three times the distance between its directrices.
$B$. Hyperbola in which the transverse axis is twice the conjugate axis.
$C$. Hyperbola with asymptotes $x+y+1=0$ and $x-y+3=0$.