If $P(x_1, y_1), Q(x_2, y_2), R(x_3, y_3)$ and $S(x_4, y_4)$ are $4$ concyclic points on the rectangular hyperbola $xy = c^2$,the coordinates of the orthocentre of the triangle $PQR$ are:

The equation of the directrices of the conic $x^2 + 2x - y^2 + 5 = 0$ are

The angle between the asymptotes of the hyperbola $x^2-3y^2=3$ is

Let the equation of two diameters of a circle $x^{2} + y^{2} - 2x + 2fy + 1 = 0$ be $2px - y = 1$ and $2x + py = 4p$. Then the slope $m \in (0, \infty)$ of the tangent to the hyperbola $3x^{2} - y^{2} = 3$ passing through the centre of the circle is equal to $......$

The locus of the centroid of the triangle formed by any point $P$ on the hyperbola $16x^{2}-9y^{2}+32x+36y-164=0$ and its foci is:

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