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

The equation of the hyperbola whose eccentricity is $\frac{5}{3}$ and the distance between the foci is $10$ units is:

The locus of the point of intersection of the straight lines $\frac{x}{a} - \frac{y}{b} = m$ and $\frac{x}{a} + \frac{y}{b} = \frac{1}{m}$ is

The equation of a tangent to the hyperbola $4x^2 - 5y^2 = 20$ parallel to the line $x - y = 2$ is

The condition that the straight line $lx + my = n$ may be a normal to the hyperbola $b^2x^2 - a^2y^2 = a^2b^2$ is given by

If the product of the slopes of the tangents drawn from an external point $P(h, k)$ to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ is a constant $k^2$,then the locus of $P$ is