If $P(\theta) = (x_1, \frac{3 \sqrt{5}}{2})$,$0 < \theta < \frac{\pi}{2}$ is a point on the hyperbola $\frac{x^2}{25} - \frac{y^2}{9} = 1$,where $\theta$ is the parameter in its parametric form,then $2 x_1 + 9 \sin^2 \theta = $

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 $TRUE$?
$(A)$ $1 < e < \sqrt{2}$
$(B)$ $\sqrt{2} < e < 2$
$(C)$ $\Delta = a^4$
$(D)$ $\Delta = b^4$

If the line $3x - my + 5 = 0$ is a tangent to the hyperbola $3x^2 - 4y^2 = 300$,then the square of the $Y$-intercept made by this tangent line is:

The equation of the tangents to the hyperbola $3x^2 - 4y^2 = 12$ which cut equal intercepts from the axes are:

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

If $(8,2)$ is a point on the hyperbola whose length of the transverse axis is $12$ and conjugate axis is $x=0$,then the eccentricity of that hyperbola is