The eccentricity of the hyperbola $2x^2 - y^2 = 6$ is

If the variable line $y = kx + 2h$ is tangent to an ellipse $2x^2 + 3y^2 = 6$,the locus of $P(h, k)$ is a conic $C$ whose eccentricity equals

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:

Let $X$-axis be the transverse axis and $Y$-axis be the conjugate axis of a hyperbola $H$. Let the eccentricity of $H$ be the reciprocal of the eccentricity of the ellipse $\frac{x^2}{4} + \frac{y^2}{2} = 1$. If $(5, 4)$ is a point on $H$,then the length of the transverse axis of $H$ is

The difference of the focal distances of any point on the hyperbola $9x^2 - 16y^2 = 144$ is

Let $A(2 \sec \theta, 3 \tan \theta)$ and $B(2 \sec \phi, 3 \tan \phi)$ where $\theta+\phi=\frac{\pi}{2}$ be two points on the hyperbola $\frac{x^2}{4}-\frac{y^2}{9}=1$. If $(\alpha, \beta)$ is the point of intersection of normals to the hyperbola at $A$ and $B$, then $\beta$ is equal to