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

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

Let the domain of the function $f(x) = \log_{3}\log_{5}\log_{7}(9x - x^{2} - 13)$ be the interval $(m, n)$. Let the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ have eccentricity $\frac{n}{3}$ and the length of the latus rectum $\frac{8m}{3}$. Then $b^{2} - a^{2}$ is equal to:

If the perimeter of a triangle is $20$ and two of its vertices are $(-5, 0)$ and $(6, 0)$,then the locus of the third vertex is:

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

The magnitude of the gradient of the tangent at an extremity of the latera recta of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ is equal to (where $e$ is the eccentricity of the hyperbola).