$$x^2-2py=0$$

The focus lies on the OY axis at the point $\left(0,\frac{p}{2}\right)$. You can modify the value of the parameter p with the slider.

Activating the tangent, you can also verify that a ray parallel to the axis is reflected in the parabola to the focus; compare the angles of incidence and reflection on the tangent at the point.

Given a point P, outside the parabola, 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.

When the point P its at the focus, then its polar line is the directrix of the parabola.

You can check that as point P moves away from the origin in a direction not parallel to the OY axis, its polar line approaches a parallel to the OY axis. However, if even moving away, you keep the point P on the OY axis, the polar line seems to "go to infinity". Why?

Polar Line

Locus