Find the slope of the tangent to the hyperbola $2x^2 - 3y^2 = 6$ at the point $(3, 2)$.

If a circle of radius $4 \text{ cm}$ passes through the foci of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{4} = 1$ and is concentric with the hyperbola,then the eccentricity of the conjugate hyperbola of that hyperbola is

Let $a$ and $b$ respectively be the semi-transverse and semi-conjugate axes of a hyperbola whose eccentricity satisfies the equation $9e^2 - 18e + 5 = 0$. If $S(5, 0)$ is a focus and $5x = 9$ is the corresponding directrix of this hyperbola,then $a^2 - b^2$ is equal to

Find the equation of the hyperbola satisfying the given conditions: Foci $(0, \pm \sqrt{10})$,passing through $(2, 3)$.

The equation of the hyperbola whose foci are $(6, 4)$ and $(-4, 4)$ and eccentricity $e = 2$ is given by

If $P(\theta)$ lies on the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$ and $S$ and $S^{\prime}$ are the foci of the hyperbola,then $SP \cdot S^{\prime}P =$