The locus of the foot of the perpendicular drawn from any focus to a variable tangent of the hyperbola $4x^2 - 9y^2 - 8x - 18y = 41$ is:

The eccentricity of the hyperbola whose length of the latus rectum is equal to $8$ and the length of its conjugate axis is equal to half of the distance between its foci is:

The equation of the tangent to the hyperbola $4y^2 = x^2 - 1$ at the point $(1, 0)$ is

Let $P(a \sec \theta, b \tan \theta)$ and $Q(a \sec \phi, b \tan \phi)$ be two points such that $\theta+\phi=\frac{\pi}{2}$ 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 at $P$ and $Q$,then $k=$

If $c$ is a real number and $\frac{x^2}{c-12}+\frac{y^2}{7-c}=1$ represents a hyperbola,then

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