$$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$$

The foci lie over the OX axis. When a=b then the hyperbola is called equilateral. In this case the asymptotes  are perpendicular.

You can modify the dimensions a and b of the hyperbola, respectively moving the points blue and red.

Activating the tangent, you can also verify that a ray coming from one focus is reflected in the hyperbola to the other; compare the angles of incidence and reflection on the tangent at the point.

Given a point P, outside the hyperbola, its polar line joins the points of tangency of the tangent lines to the conic that pass through P. Specifically, if the point P belongs to the conic, the polar line coincides with the tangent at P.

The diameters of the hyperbola are the polar lines of the points of infinity. We know that in the case of the ellipse, all lines passing through the center are diameters.

You can check that as point P moves away from the origin, its polar line gets closer to the center.

Directrices 

Polar Line

Locus