If the eccentricity of the standard hyperbola passing through the point $(4, 6)$ is $2$,then the equation of the tangent to the hyperbola at $(4, 6)$ is

If the normal to the rectangular hyperbola $x^2-y^2=1$ at the point $P$ with parameter $\theta_1 = \frac{\pi}{4}$ meets the curve again at $Q$ with parameter $\theta_2$,then find the value of $\sec^2 \theta_2 + \tan \theta_2$.

The equation of the line passing through the points $\left(ct_1, \frac{c}{t_1}\right)$ and $\left(ct_2, \frac{c}{t_2}\right)$ is

Find the equation of the hyperbola satisfying the given conditions: Vertices $(\pm 7, 0)$,$e = \frac{4}{3}$.

If a circle intersects the hyperbola $xy = 1$ at four points $(x_r, y_r)$ for $r = 1, 2, 3, 4$,then:

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 $k=$