The eccentricity of the hyperbola $4x^2 - 9y^2 = 36$ is

The values of $m$ for which the line $y=mx+2$ is a tangent to the hyperbola $4x^2-9y^2=36$ are

If the equation of the hyperbola having $(8,3)$ and $(0,3)$ as foci and $\frac{4}{3}$ as eccentricity is $\frac{(x-\alpha)^2}{p}-\frac{(y-\beta)^2}{q}=1$,then $p+q=$

If the eccentricity of a hyperbola is $\frac{5}{3}$,then the eccentricity of its conjugate hyperbola is

Let $P, Q, R, S$ be the points of intersection of the circle $x^2+y^2=4$ and the hyperbola $xy=\sqrt{3}$. If $P=(\alpha, \beta)$ and $\alpha>\beta>0$,then the equation of the tangent drawn at $P$ to the hyperbola is

Let $a$ and $b$ be positive real numbers such that $a > 1$ and $b < a$. Let $P$ be a point in the first quadrant that lies on the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$. Suppose the tangent to the hyperbola at $P$ passes through the point $(1, 0)$,and suppose the normal to the hyperbola at $P$ cuts off equal intercepts on the coordinate axes. Let $\Delta$ denote the area of the triangle formed by the tangent at $P$,the normal at $P$,and the $x$-axis. If $e$ denotes the eccentricity of the hyperbola,then which of the following statements is/are $TRUE$?
$(A)$ $1 < e < \sqrt{2}$
$(B)$ $\sqrt{2} < e < 2$
$(C)$ $\Delta = a^4$
$(D)$ $\Delta = b^4$