Q.A circle whose centre is $(4,-1)$ passes through the focus of the parabola $x^2 + 16y = 0$. Show that the circle touches the directrix of the parabola.

Starting from the focus-directrix property with $e=1$ (so $SP=PM$ exactly), and choosing the midpoint of the perpendicular from focus to directrix as the origin, the algebra collapses beautifully to the standard equation of a parabola, $y^2=4ax$ (where $a>0$ is the distance from the vertex to the focus).

From this one equation, every other property follows by direct calculation: the parabola is symmetric about its axis, lies entirely on one side of the vertex, and extends to infinity in one direction. Its focus sits at $(a,0)$, its directrix is the line $x=-a$, and — very usefully — the focal distance of any point $(x_1,y_1)$ on the curve is simply $x_1+a$ (no distance formula needed). The latus rectum (the focal chord perpendicular to the axis) has length $4a$, with end points $(a,\pm2a)$. …