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 $m_1$ and $m_2$ be the slopes of the tangents drawn from the point $P(4, 1)$ to the hyperbola $H: \frac{y^2}{25} - \frac{x^2}{16} = 1$. If $Q$ is the point from which the tangents drawn to $H$ have slopes $|m_1|$ and $|m_2|$ and they make positive intercepts $\alpha$ and $\beta$ on the $x$-axis,then $\frac{(PQ)^2}{\alpha \beta}$ is equal to $............$.

The length of the latus rectum of the curve $xy = 7x + 5y$ is

If the line $y = mx + c$ is a tangent to the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$,then the locus of the point $P(m, c)$ is:

Let the origin be the centre,$(\pm 3, 0)$ be the foci,and $\frac{3}{2}$ be the eccentricity of a hyperbola. Then the line $2x - y - 1 = 0$

Let $P(a \sec \theta, b \tan \theta)$ and $Q(a \sec \phi, b \tan \phi)$ where $\theta + \phi = \frac{\pi}{2}$ be two points on the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$. If $(h, k)$ is the point of intersection of the normals drawn at $P$ and $Q$,then $k=$