Equal distance to a point and a line — and the one number that unites all three.
Loop a string around two pins and the constant sum of distances drew an ellipse; hold the difference fixed and two hyperbola branches flew apart. One pattern is left: keep a point exactly as far from a single point as from a single line, and the trail is a parabola — the arc of a fountain, the dish of a satellite antenna, the path of a thrown ball. Then comes the payoff of the whole stage. Slice one cone at a growing tilt and out fall a circle, an ellipse, a parabola, a hyperbola — and a single number, the eccentricity e, decides which one you get. Geometry has been keeping its books with algebra all along; here the ledger balances.
Fix one point, the focus F, and one straight line, the directrix. Now collect every point P that is equally far from both — the distance to F exactly matches the perpendicular distance to the directrix. Those points do not scatter; they join into one smooth, open arc. That arc is the parabola.
Why does it open and never close? Far from F, the perpendicular gap to a straight line grows just as fast as the straight-line gap to a point, so the two distances can keep matching forever — the curve runs off to infinity instead of looping back. Compare it with the ellipse (a bounded loop) and the hyperbola (two unbounded branches): the parabola is the single open arc that sits exactly between them.
A parabola is the set of points equidistant from a fixed point (the focus) and a fixed line (the directrix): |PF| = distance from P to the directrix. One focus, one directrix — and that single rule fixes the entire curve.
Set the curve in its tidiest pose: vertex at the origin, opening to the right. Put the focus at F(p/2, 0) and the directrix at the vertical line x = −p/2, with p > 0. A point P(x, y) sits on the parabola when its two distances agree:
|PF| = √((x − p/2)2 + y2) = x + p/2 = distance to the directrix.
Square both sides and the cross terms cancel cleanly: (x − p/2)2 + y2 = (x + p/2)2, which collapses to the famous one-liner
y2 = 2px — focus (p/2, 0), directrix x = −p/2.
The number p is the distance from the focus to the directrix; a larger p pushes them apart and the parabola opens wider. Three siblings share the same shape in different poses: y2 = 2px opens right, y2 = −2px opens left, and x2 = 2py opens up.
The familiar y = ax2 from quadratic functions is this very curve, just standing up. Rewrite it as x2 = 1ay, which is x2 = 2py with 2p = 1/a. Same parabola, rotated a quarter turn — a sideways y2 = 2px versus an upright y = ax2.
Slide the point P along the parabola and watch the green focal radius |PF| and the slate drop to the directrix stay equal — that equality is the curve. The stepper for p opens it wider or narrower.
The focus and the directrix bracket the curve: the focus sits inside the cup, the directrix lies outside it, and the vertex is the one point exactly halfway between them. That layout gives the parabola its most useful trick — the reflective property.
Aim a beam of parallel rays straight into the cup, along the axis. Each ray bounces off the curve and — every single one — passes through the focus. Run it backwards and a lamp at the focus throws out a perfectly parallel beam. That is why a satellite dish gathers faint signals to one receiver at the focus, why a headlight reflector and a solar-cooker concentrate light, and why a radio telescope is a parabola. One point collects what an entire open mouth catches.
A parabola has one focus and one directrix — not two foci like the ellipse and hyperbola. And mind the pose: y2 = 2px opens sideways, while y = ax2 opens up. Same shape, different orientation — read the equation, not your habit.
Now stand back from the three curves and take a single cone — a double cone, two cups tip to tip. Slice it with a flat plane and watch the cross-section change as you tilt the knife:
That is why these four are called the conic sections: one cone, four slices. The Greeks studied them as pure geometry two thousand years ago; Descartes' grid (the start of this whole stage) turned each slice into an equation you can compute.
Here is the deep reason the four curves are siblings. Go back to a focus and a directrix, and for any point P form one ratio:
e = distance from P to the focusdistance from P to the directrix (the eccentricity).
Demand that this ratio be the same at every point of the curve, and the value of e alone decides the shape:
| eccentricity | curve | character |
|---|---|---|
| e < 1 | ellipse | bounded; focus pulls P closer than the directrix |
| e = 1 | parabola | the knife-edge; distances exactly equal |
| e > 1 | hyperbola | unbounded; the directrix wins, branches fly apart |
A circle is the limiting case e = 0 (the focus is the centre, the directrix infinitely far). So one definition — a focus, a directrix, and the ratio e — sweeps out every conic. The parabola at e = 1 is the precise boundary between the bounded ellipse and the unbounded hyperbola.
Slide e from small up past 1. Below 1 the curve is a closed ellipse; at exactly 1 it springs open into a parabola; above 1 it splits into a two-branch hyperbola. One number, three curves.
Look at what the coordinate method did across this whole stage. A point became a pair (x, y). A distance became a square root. A circle, an ellipse, a hyperbola, a parabola — each became an equation you could solve, intersect, and measure. Shape turned into symbol, and symbol gave answers back as shape. That round-trip is the heart of analytic geometry, Descartes' great gift.
It even hints at what comes next. To find where a line meets a circle we solved a quadratic — and sometimes the discriminant came out negative. Geometrically: the line misses the circle, no crossing point. But algebra still wants a √ of a negative number. What if we let it have one? That single question — what is √(−1)? — opens a brand-new axis at right angles to the real one, and with it the world of Stage 31 · Complex Numbers. The coordinate plane you have been drawing on is about to grow a second, imaginary dimension.
| curve | defining rule | eccentricity | foci | a, b, c |
|---|---|---|---|---|
| ellipse | |PF₁| + |PF₂| = 2a (constant sum) | e < 1 | two | a2 = b2 + c2 |
| parabola | |PF| = dist. to directrix (equal) | e = 1 | one | y2 = 2px |
| hyperbola | | |PF₁| − |PF₂| | = 2a (constant difference) | e > 1 | two | c2 = a2 + b2 |
Parabola facts: vertex at the origin, y2 = 2px opens right with focus (p/2, 0) and directrix x = −p/2; p is the focus-to-directrix distance and a larger p opens it wider. The four conics are one cone sliced four ways, threaded by the single number e — and the coordinate method now leads on to Stage 31 · Complex Numbers.
A parabola has equation y2 = 8x. Find its focus and directrix.
Match y2 = 2px: 2p = 8, so p = 4. Focus = (p/2, 0) = (2, 0); directrix x = −p/2 = x = −2.
For the parabola y2 = 6x, the point P lies on the curve with x-coordinate 4. Show that |PF| equals the distance from P to the directrix.
Here 2p = 6 so p = 3; focus (1.5, 0), directrix x = −1.5. At x = 4: y2 = 24, so P = (4, ±√24). Distance to directrix = 4 − (−1.5) = 5.5. |PF| = √((4 − 1.5)2 + 24) = √(6.25 + 24) = √30.25 = 5.5. They match — exactly the focus–directrix property.
Rewrite y = 12x2 in the form x2 = 2py and state the focus.
y = ½x2 ⟺ x2 = 2y, so 2p = 2 and p = 1. This parabola opens up with focus (0, p/2) = (0, ½) and directrix y = −½. Same shape as a sideways y2 = 2px, just standing up.
A conic is defined by the focus–directrix ratio e. Name the curve for e = 0.5, for e = 1, and for e = 2.
e = 0.5 < 1 → ellipse; e = 1 → parabola (the knife-edge); e = 2 > 1 → hyperbola. The single ratio e sorts all three.
You tilt a plane through a cone, starting horizontal and tipping steeper. List, in order, the cross-sections you sweep through.
Circle → ellipse → parabola → hyperbola. The circle is the level cut; tilting gives an ever-longer ellipse; the parabola appears at the exact tilt parallel to the cone's slant side; tilt past that and the plane catches both cups, giving the two-branch hyperbola.
Explain in one or two sentences why a satellite dish is shaped as a parabola.
By the reflective property, every ray that arrives parallel to the axis reflects off the parabola straight to the focus. So the dish concentrates a whole sheet of faint incoming signal onto one receiver placed at the focus — maximum signal, one detector.
Six questions to lock it in. Tap the answer you think is right.
This closing lesson develops the parabola from its focus–directrix definition and then unifies all four conic sections through eccentricity, matching HSG-GPE.A.2 (derive the equation of a parabola given a focus and directrix), HSG-GPE.A.3 (ellipse and hyperbola from the sum/difference of focal distances), and HSG-GPE.B.4 (use coordinates to prove geometric facts algebraically). The algebra of squaring the distance equation and the line–conic intersection that motivates Stage 31 draw on HSA-REI.B.4 (solving quadratics, including the discriminant) and HSA-REI.C.7 (solving a linear–quadratic system). Every distance, focus, and eccentricity in the figures is computed by the lesson's geometry engine, so each picture is numerically faithful — encourage learners to read the live readouts and confirm |PF| really does equal the distance to the directrix at every point.