The equation $3x^2 + 7xy + 2y^2 + 5x + 5y + 2 = 0$ represents:

Statement $ (A) $: The point $(5, -4)$ lies inside the hyperbola $y^2 - 9x^2 + 1 = 0$.
Reason $(R)$: $A$ point $(x_1, y_1)$ lies inside the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ if $\frac{x_1^2}{a^2} - \frac{y_1^2}{b^2} - 1 < 0$.

The equation of the director circle of the hyperbola $\frac{x^2}{16} - \frac{y^2}{4} = 1$ is given by

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

Consider the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ having one of its foci at $P(-3,0)$. If the latus rectum through its other focus subtends a right angle at $P$ and $a^2b^2 = \alpha\sqrt{2} - \beta$,where $\alpha, \beta \in N$,then find the value of $\alpha + \beta$.

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