If $A(4,0)$ and $B(-4,0)$ are two points,then the locus of a point $P$ such that $PA - PB = 4$ is

The equation of the normal to the hyperbola $\frac{x^2}{16} - \frac{y^2}{9} = 1$ at $(-4, 0)$ is

Equation of one of the tangents passing through $(2,8)$ to the hyperbola $5 x^2-y^2=5$ is

The equation of the directrices of the hyperbola $3x^{2}-3y^{2}-18x+12y+2=0$ is

Let the eccentricity of the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$ be reciprocal to that of the ellipse $x^{2}+9y^{2}=9$. Then the ratio $a^{2}:b^{2}$ equals:

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