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 $TRUE$?

$(A)$ $1 < e < \sqrt{2}$

$(B)$ $\sqrt{2} < e < 2$

$(C)$ $\Delta=a^4$

$(D)$ $\Delta=b^4$

A hyperbola passes through the points $(3, 2)$ and $(-17, 12)$ and has its centre at origin and transverse axis is along $x$ - axis. The length of its transverse axis is

The length of the latus rectum and directrices of a hyperbola with eccentricity e are 9 and $\mathrm{x}= \pm \frac{4}{\sqrt{3}}$, respectively. Let the line $y-\sqrt{3} \mathrm{x}+\sqrt{3}=0$ touch this hyperbola at $\left(\mathrm{x}_0, \mathrm{y}_0\right)$. If $\mathrm{m}$ is the product of the focal distances of the point $\left(\mathrm{x}_0, \mathrm{y}_0\right)$, then $4 \mathrm{e}^2+\mathrm{m}$ is equal to ...........

Length of latusrectum of curve $xy = 7x + 5y$ is

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Let the foci of a hyperbola $\mathrm{H}$ coincide with the foci of the ellipse $E: \frac{(x-1)^2}{100}+\frac{(y-1)^2}{75}=1$ and the eccentricity of the hyperbola $\mathrm{H}$ be the reciprocal of the eccentricity of the ellipse $E$. If the length of the transverse axis of $\mathrm{H}$ is $\alpha$ and the length of its conjugate axis is $\beta$, then $3 \alpha^2+2 \beta^2$ is equal to :