The eccentricity of the curve $x^2 - y^2 = a^2$ is

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

If $e_1$ and $e_2$ are respectively the eccentricities of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ and its conjugate hyperbola,then the line $\frac{x}{2 e_1}+\frac{y}{2 e_2}=1$ touches the circle having centre at the origin. Find its radius.

The distance between the foci of a hyperbola is $16$ and its eccentricity is $\sqrt{2}$. Its equation is

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 $PQ$ be a chord of the hyperbola $\frac{x^2}{4} - \frac{y^2}{b^2} = 1$,perpendicular to the $x$-axis such that $OPQ$ is an equilateral triangle,where $O$ is the centre of the hyperbola. If the eccentricity of the hyperbola is $\sqrt{3}$,then the area of the triangle $OPQ$ is: