The equations of the transverse and conjugate axes of the hyperbola $16x^2 - y^2 + 64x + 4y + 44 = 0$ are

If $\frac{x}{a} + \frac{y}{b} = 1$ is a tangent to the curve $x = Kt, y = \frac{K}{t}, K > 0$,then

The asymptotes of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$,with any tangent to the hyperbola form a triangle whose area is $a^2 \tan (\alpha)$. Then its eccentricity equals

The number of normals that can be drawn from an external point to the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ is:

Circles are drawn on chords of the rectangular hyperbola $xy = c^2$ parallel to the line $y = x$ as diameters. All such circles pass through two fixed points whose coordinates are:

The product of the lengths of the perpendiculars drawn from the foci to any tangent to the hyperbola $x^{2} - \frac{y^{2}}{4} = 1$ is: