Parabola — standard forms (4 orientations), focus, directrix, latus rectum, axis: Visual walkthrough

Step 1 — Put down the two ingredients

WHAT. A parabola is made from exactly two things: a fixed point and a fixed straight line. We call the point the focus and the line the directrix. That is all the raw material.

WHY. Before any algebra, we must be able to see the objects the algebra talks about. Every symbol later ( $a$ , $x$ , $y$ ) will be measured against these two things — so they come first.

PICTURE. The yellow dot is the focus $F$ . The red vertical line is the directrix $ℓ$ . Nothing is a "parabola" yet — we only have the ingredients.

Figure — Parabola — standard forms (4 orientations), focus, directrix, latus rectum, axis

Step 2 — Choose smart coordinates (why we get to pick)

WHAT. We drop a grid on the picture. We place the midpoint between the point and the line at the origin $(0, 0)$ . We aim the horizontal axis (the $x$ -axis) straight from the line, through that midpoint, into the point. Then we give the point-to-midpoint distance a name: the letter $a$ , taken positive.

So the focus lands at $F = (a, 0)$ and — because the midpoint is halfway — the directrix is the vertical line $x = - a$ .

WHY. The parabola exists no matter where we draw axes, but a lazy choice of axes makes ugly algebra. Centring on the midpoint and pointing the axis along the symmetry line makes every term below cancel beautifully. This is allowed: we are only labelling space, not changing the shape.

PICTURE. Notice the two equal green gaps of length $a$ : origin-to-focus, and origin-to-directrix. The midpoint (origin) is the future vertex.

Step 3 — Meet a moving point $P$ and its two distances

WHAT. Pick any point $P = (x, y)$ in the plane. It has two distances that matter:

its distance to the focus, written $P F$ ;
its distance to the directrix, written $P M$ (the shortest, i.e. perpendicular, hop to the line).

WHY. The rule from Step 1 compares exactly these two lengths. To turn the rule into an equation we must first write each length using $x$ and $y$ . The tool for length-between-points is the distance formula — we use it precisely because it converts a geometric distance into an algebraic expression we can manipulate.

PICTURE. Blue segment $= P F$ . Red dashed segment $= P M$ , dropped straight onto the directrix. Right now they are different lengths — $P$ is a general point, not yet on the parabola.

Step 4 — Impose the rule and kill the roots

WHAT. Now demand the two distances be equal (Step 1's rule), then square both sides.

$(x - a)^{2} + y^{2} = ∣ x + a ∣ ⟹ (x - a)^{2} + y^{2} = (x + a)^{2}$

WHY square? A square root and an absolute value are both awkward to expand. But both sides here are non-negative (a distance can't be negative), and for non-negative numbers "equal" and "squares equal" mean the same thing. Squaring is the surgical move that deletes the and the $∣ ∣$ in one stroke, losing no solutions.

PICTURE. This is the frame where the blue segment and the red dashed segment finally have the same length — $P$ has snapped onto the curve. Watch the two lengths become one.

Step 5 — Expand, cancel, and read off the answer

WHAT. Multiply out both squared brackets and see what survives.

$(x - a)^{2} x^{2} - 2 a x + a^{2} + y^{2} = (x + a)^{2} x^{2} + 2 a x + a^{2}$

The $x^{2}$ appears on both sides — cancel it. The $a^{2}$ appears on both sides — cancel it. What is left:

$y^{2} = 2 a x - (- 2 a x) = 4 a x$

WHY it cancels so cleanly. This is the payoff of Step 2. Centring on the midpoint made the two brackets differ only in the sign of the middle term $\pm 2 a x$ ; everything symmetric ( $x^{2}$ , $a^{2}$ ) matched and vanished, leaving one term standing.

PICTURE. The two expansions laid side by side; matching terms are struck through, and the lone surviving $4 a x$ is highlighted — the equation of the curve.

Step 6 — Cash out the latus rectum (the "width at the focus")

WHAT. Slice the parabola with the vertical line through the focus, $x = a$ . The chord you cut is the latus rectum. Find its length.

Substitute $x = a$ into $y^{2} = 4 a x$ : $y^{2} = 4 a \cdot a = 4 a^{2} ⟹ y = \pm 2 a .$ Endpoints $(a, 2 a)$ and $(a, - 2 a)$ ; length $= 2 a - (- 2 a) = 4a$ .

WHY care. With one number, $4 a$ , you know how wide the parabola is at its focus — enough to sketch it fast. It's also why we wrote the coefficient as " $4 a$ " and not some plain constant: it reads the width off directly.

PICTURE. The green vertical chord at $x = a$ , its two endpoints marked, its total length labelled $4 a$ .

Step 7 — The degenerate & limiting cases (never left in the dark)

WHAT. We check the extremes so no scenario surprises you.

$a \to 0^{+}$ (focus meets the line): the two green gaps in Step 2 shrink to nothing, $y^{2} = 0$ , so the "parabola" collapses to the single line $y = 0$ — a degenerate case. This is exactly why the definition forbids the directrix passing through the focus.
$a$ large: the $4 a$ chord is huge, so the curve flares wide and opens lazily.
$a$ small (but $> 0$ ): narrow, tight curve hugging its axis.
The vertex $(0, 0)$ : plug $x = 0$ : $y^{2} = 0$ , one point — the parabola touches its axis exactly once, at the vertex, the closest approach to both focus and directrix.
Sign of $x$ : since $y^{2} \geq 0$ and $4 a > 0$ , we need $x \geq 0$ . The curve lives only to the right — it never appears where $x < 0$ . That is the algebra telling you it opens rightward.

WHY. A derivation you can't stress-test isn't finished. These edges show where the curve can and cannot go and why the definition's fine print exists.

PICTURE. Three parabolas sharing a vertex: small $a$ (narrow), medium, large $a$ (wide), plus the collapsed line at $a = 0$ .

The one-picture summary

Everything at once. Two ingredients (yellow focus, red directrix) → equal blue/red distances for a moving $P$ → square & cancel → $y^{2} = 4 a x$ → width $4 a$ at the focus. Trace the picture and you have re-derived the whole topic.

Recall Feynman retelling — say it like a story

Put a dot and a straight fence on the ground, the dot not on the fence. Stand where the fence is as far away as the dot — same distance both ways. Every such spot, joined up, is a parabola. To pin it with numbers, stand exactly halfway between dot and fence and call that the middle (the vertex); call the halfway distance $a$ . Now write "my distance to the dot equals my distance to the fence," turn both distances into $x$ -and- $y$ using the straight-line distance rule, and square to lose the roots. The messy parts ( $x^{2}$ , $a^{2}$ ) match on both sides and disappear, leaving the clean $y^{2} = 4 a x$ . Slice through the dot straight up-and-down and the curve is $4 a$ wide there. If the dot were on the fence ( $a = 0$ ) the whole thing squashes to a line — which is why the rule bans that. And because a square is never negative, the curve only ever reaches out to the right.

Connections

Parent topic — the four orientations & worked examples
Coordinate Geometry — Distance Formula — the single tool used in Steps 3–5
Eccentricity and focus-directrix definition — the parabola is the $e = 1$ member
Conic Sections — Overview, Ellipse — standard forms, Hyperbola — standard forms — compare the other conics
Tangents and Normals to a Parabola — what to do once you have $y^{2} = 4 a x$
Projectile Motion — parabolas in the physical world