The product of the lengths of perpendiculars drawn from any point on the hyperbola $x^2 - 2y^2 - 2 = 0$ to its asymptotes is:

If $e$ and $e'$ are the eccentricities of a hyperbola and its conjugate hyperbola respectively,then:

On a rectangular hyperbola $x^2-y^2=a^2, a > 0$,three points $A, B, C$ are taken as follows: $A=(-a, 0)$; $B$ and $C$ are placed symmetrically with respect to the $X$-axis on the branch of the hyperbola not containing $A$. Suppose that the $\triangle ABC$ is equilateral. If the side length of the $\triangle ABC$ is $ka$,then $k$ lies in the interval

The length of the latus rectum of the hyperbola $9x^2 - 16y^2 - 18x - 32y - 151 = 0$ is

If the circle $x^2 + y^2 = a^2$ intersects the hyperbola $xy = c^2$ in four points $P(x_1, y_1), Q(x_2, y_2), R(x_3, y_3), S(x_4, y_4)$,then:

Two points $A$ and $B$ have coordinates $(1, 0)$ and $(-1, 0)$ respectively,and $Q$ is a point which satisfies the relation $AQ - BQ = \pm 1$. The locus of $Q$ is: