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 tangent and normal to a rectangular hyperbola $xy = c^2$ at a variable point cut off intercept  $a_1, a_2$ on $x-$ axis and $b_1, b_2$ on $y-$ axis, then $(a_1a_2 + b_1b_2)$ is

Area of the quadrilateral formed with the foci of the hyperbola $\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} = 1$ and $\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} =  - 1$ is

Find the equation of the hyperbola satisfying the give conditions: Foci $(\pm 3 \sqrt{5},\,0),$ the latus rectum is of length $8$

An ellipse $E: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ passes through the vertices of the hyperbola $H: \frac{x^{2}}{49}-\frac{y^{2}}{64}=-1$. Let the major and minor axes of the ellipse $E$ coincide with the transverse and conjugate axes of the hyperbola $H$. Let the product of the eccentricities of $E$ and $H$ be $\frac{1}{2}$. If $l$ is the length of the latus rectum of the ellipse $E$, then the value of $113\,l$ is equal to $....$

At the point of intersection of the rectangular hyperbola $ xy = c^2 $ and the parabola $y^2 = 4ax$  tangents to the rectangular hyperbola and the parabola make an angle $ \theta $ and $ \phi $ respectively with the axis of $X$, then