Q.For each of the following parabolas, find focus, equation of the directrix, length of the latus rectum, and ends of the latus rectum: $5x^2 = 24y$.

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 $(a,\pm2a)$.

Because a parabola can open in any of the four axis-aligned directions, there are four standard forms in total: $y^2=4ax$ (opens right), $y^2=-4ax$ (opens left), $x^2=4by$ (opens up), and $x^2=-4by$ (opens down) — each with its own mirrored version of the focus/directrix/latus-rectum formulas above. When the vertex is shifted away from the origin to some point $(h,k)$ while the axis stays parallel to a coordinate axis, the equation becomes (for a horizontal axis) $(y-k)^2=4a(x-h)$; completing the square on any expanded quadratic-in-one-variable equation recovers this shifted standard form, from which the vertex, focus and directrix can all be read off after translating back by $(h,k)$.