The eccentricity of the conic ${x^2} - 4{y^2} = 1$ is

The equation $13[{(x - 1)^2} + {(y - 2)^2}] = 3{(2x + 3y - 2)^2}$ represents

If the center,vertex,and focus of a hyperbola are $(0, 0)$,$(4, 0)$,and $(6, 0)$ respectively,then the equation of the hyperbola is:

If $l$ is the maximum value of $-3x^2+4x+1$ and $m$ is the minimum value of $3x^2+4x+1$,then the equation of the hyperbola having foci at $(l, 0)$ and $(7m, 0)$ and eccentricity $e=2$ is

The equation $x^2 - 4y^2 - 2x + 16y - 40 = 0$ represents:

Let $S$ be the focus of the hyperbola $\frac{x^2}{3}-\frac{y^2}{5}=1$,on the positive $x$-axis. Let $C$ be the circle with its centre at $A(\sqrt{6}, \sqrt{5})$ and passing through the point $S$. If $O$ is the origin and $SAB$ is a diameter of $C$,then the square of the area of the triangle $OSB$ is equal to ....................