The transverse axis of a hyperbola is along the $x$-axis and its length is $2a$. The vertex of the hyperbola bisects the line segment joining the centre and the focus. The equation of the hyperbola is

Let $e$ be the eccentricity of the ellipse $\frac{x^2}{4}+\frac{y^2}{9}=1$. If $\frac{1}{e}$ is the eccentricity of a hyperbola,then the eccentricity of its conjugate hyperbola is

Let $P (3 \sec \theta, 2 \tan \theta)$ and $Q (3 \sec \phi, 2 \tan \phi)$ where $\theta + \phi = \frac{\pi}{2}$,be two distinct points on the hyperbola $\frac{x^2}{9} - \frac{y^2}{4} = 1$. Then the ordinate of the point of intersection of the normals at $P$ and $Q$ is

When is the line $ℓx + my + n = 0$ a tangent to the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$?

If $lx + my = 1$ is a normal to the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$,then $a^2 m^2 - b^2 l^2 =$

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