The equation of the tangent to the conic ${x^2} - {y^2} - 8x + 2y + 11 = 0$ at $(2, 1)$ is

Let $(1, 2)$ be the focus and $x+y+1=0$ be the directrix of a hyperbola $H$. If $\sqrt{3}$ is the eccentricity of $H$,then its equation is

Find the equation of the hyperbola whose foci are $(0, \pm 12)$ and the length of the latus rectum is $36$.

The locus of a point which moves such that the difference of its distances from two fixed points is always a constant is

If the eccentricity of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ is $\sec \alpha$,then the area of the triangle formed by the asymptotes of the hyperbola with any of its tangent is

The locus of the point of intersection of the lines $\frac{x}{a} + \frac{y}{b} = \lambda$ and $\frac{x}{a} - \frac{y}{b} = \frac{1}{\lambda}$ (where $\lambda$ is a parameter) is: