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 $x y = c^2$ , the co-ordinates of the orthocentre of the triangle $ PQR$  are :

The locus of the midpoints of the chord of the circle, $x^{2}+y^{2}=25$ which is tangent to the hyperbola $, \frac{ x ^{2}}{9}-\frac{ y ^{2}}{16}=1$ is

If a hyperbola has length of its conjugate axis equal to $5$ and the distance between its foci is $13$, then the eccentricity of the hyperbola is

Consider a branch of the hyperbola $x^2-2 y^2-2 \sqrt{2} x-4 \sqrt{2} y-6=0$ with vertex at the point $A$. Let $B$ be one of the end points of its latus rectum. If $\mathrm{C}$ is the focus of the hyperbola nearest to the point $\mathrm{A}$, then the area of the triangle $\mathrm{ABC}$ is

A common tangent $T$ to the curves $C_{1}: \frac{x^{2}}{4}+\frac{y^{2}}{9}=1$ and $C_{2}: \frac{x^{2}}{42}-\frac{y^{2}}{143}=1$ does not pass through the fourth quadrant. If $T$ touches $C _{1}$ at ( $\left.x _{1}, y _{1}\right)$ and $C _{2}$ at $\left( x _{2}, y _{2}\right)$, then $\left|2 x _{1}+ x _{2}\right|$ is equal to $……$