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 foci of the hyperbola $9{x^2} - 16{y^2} = 144$ are

If the eccentricity of the standard hyperbola passing, through the point $(4, 6)$ is $2$, then the equation of the tangent to the hyperbola at $(4, 6)$ is

The eccentricity of the conjugate hyperbola of the hyperbola ${x^2} - 3{y^2} = 1$, is

Let $a>0, b>0$. Let $e$ and $\ell$ respectively be the eccentricity and length of the latus rectum of the hyperbola $\frac{ x ^{2}}{ a ^{2}}-\frac{ y ^{2}}{ b ^{2}}=1$. Let $e ^{\prime}$ and $\ell^{\prime}$ respectively the eccentricity and length of the latus rectum of its conjugate hyperbola. If $e ^{2}=\frac{11}{14} \ell$ and $\left( e ^{\prime}\right)^{2}=\frac{11}{8} \ell^{\prime}$, then the value of $77 a+44 b$ is equal to

The coordinates of the foci of the rectangular hyperbola $xy = {c^2}$ are