If angle between asymptotes of hyperbola $\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{3} = 4$ is $\frac{\pi }{3}$, then its conjugate hyperbola is

The equation of the common tangents to the two hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ and $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ are-

Area of the quadrilateral formed with the foci of the hyperbola $\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} = 1$ and $\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} =  - 1$ is

The equation of the tangent parallel to $y - x + 5 = 0$ drawn to $\frac{{{x^2}}}{3} - \frac{{{y^2}}}{2} = 1$ is

The graph of the conic $x^2-(y-1)^2=1$ has one tangent line with positive slope that passes through the origin. The point of the tangency being $(a, b)$ then find the value of $\sin ^{-1}\left(\frac{a}{b}\right)$

The distance between the foci of a hyperbola is double the distance between its vertices and the length of its conjugate axis is $6$. The equation of the hyperbola referred to its axes as axes of co-ordinates is