The equation of the hyperbola in the standard form (with transverse axis along the $x$ -  axis) having the length of the latus rectum = $9$ units and eccentricity = $5/4$ is

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

Let $e_{1}$ and $e_{2}$ be the eccentricities of the ellipse, $\frac{x^{2}}{25}+\frac{y^{2}}{b^{2}}=1(b<5)$ and the hyperbola $\frac{ x ^{2}}{16}-\frac{ y ^{2}}{ b ^{2}}=1$ respectively satisfying $e _{1} e _{2}=1 .$ If $\alpha$ and $\beta$ are the distances between the foci of the ellipse and the foci of the hyperbola respectively, then the ordered pair $(\alpha, \beta)$ is equal to

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 product of the perpendiculars drawn from any point on a hyperbola to its asymptotes is

The locus of the point of intersection of the lines $bxt - ayt = ab$ and $bx + ay = abt$ is