Let the tangent drawn to the parabola $y ^{2}=24 x$ at the point $(\alpha, \beta)$ is perpendicular to the line $2 x$ $+2 y=5$. Then the normal to the hyperbola $\frac{x^{2}}{\alpha^{2}}-\frac{y^{2}}{\beta^{2}}=1$ at the point $(\alpha+4, \beta+4)$ does $NOT$ pass through the point.

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 $TRUE$?

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

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

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

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

The difference of the focal distance of any point on the hyperbola $9{x^2} – 16{y^2} = 144$, is

What will be equation of that chord of hyperbola $25{x^2} – 16{y^2} = 400$, whose mid point is $(5, 3)$

The equation of the normal to the hyperbola $\frac{{{x^2}}}{{16}} – \frac{{{y^2}}}{9} = 1$ at the point $(8,\;3\sqrt 3 )$ is