Parabola — standard forms (4 orientations), focus, directrix, latus rectum, axis: Question bank

Before we start, here is all the machinery the questions lean on, each item built from plain words so no symbol appears unexplained. Keep this labelled picture in front of you — every question below points back to it:

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

Definition The five things every question leans on

Focus $F$ : a fixed point. Directrix: a fixed line. A point $P$ is on the parabola exactly when its distance to $F$ equals its perpendicular distance to the directrix.
Vertex: the point of the parabola lying on the axis, exactly midway between the focus and the directrix. It is the "turning point" where the two arms meet, and it is the closest point of the curve to the focus.
$a$ : a positive length — the distance from the vertex to the focus (equal to the vertex-to-directrix distance). In $y^{2} = 4 a x$ the whole chunk $4 a$ is the coefficient of the linear term, and $4 a$ is also the latus rectum length. So $a$ is one-quarter of the coefficient — never the coefficient itself.
Axis of symmetry: the line through $F$ perpendicular to the directrix. Why perpendicular? Reflecting the whole picture must map the curve onto itself; the only mirror line that sends the directrix onto itself and keeps the focus fixed is the line through $F$ at right angles to the directrix — so that line is forced to be the axis. Consequence: if the squared variable is $y$ , the axis is the $x$ -axis (horizontal opening); if the squared variable is $x$ , the axis is the $y$ -axis (vertical opening).
Eccentricity $e$ : the ratio (distance from a point to the focus) $\div$ (distance from that point to the directrix). For a parabola this ratio is exactly $1$ at every point — that is precisely the equidistance definition rewritten as a ratio.

True or false — justify

True or false: A parabola always has exactly one axis of symmetry.

True — the axis is the unique line through the focus perpendicular to the directrix; reflecting across it swaps the two "arms" but fixes the curve. No other line does this.

True or false: The vertex is the point of the parabola closest to the focus.

True — the vertex sits midway between focus and directrix, at distance

a

from each; every other point on the curve is farther from the focus, so the vertex is the nearest point.

True or false: In

y^{2} = 4 a x

with

a > 0

, the value

a

can be read directly as the coefficient of

x

False — the coefficient of

x

4 a

, so

a

is one-quarter of it. In

y^{2} = 12 x

a = 3

, not

12

True or false: The directrix can pass through the focus.

False — by definition the directrix must not pass through the focus; if it did, "distance to focus = distance to line" would collapse and no proper parabola exists.

True or false: The eccentricity of every parabola is

1

, regardless of how wide or narrow it looks.

True — width is set by

a

(how far focus and directrix are apart), but eccentricity is the ratio distance-to-focus

\div

distance-to-directrix, which equals

1

at every point by the very definition of a parabola.

True or false:

y^{2} = 4 a x

and

y^{2} = - 4 a x

are reflections of each other across the

y

-axis.

True — replacing

x

- x

flips the curve left–right, sending the right-opening parabola to a left-opening one with the same vertex and latus rectum.

True or false: A larger

a

makes the parabola narrower.

False — a larger

a

means the focus is farther from the vertex and the latus rectum

4 a

is longer, so the parabola is wider, not narrower.

True or false: The latus rectum is the shortest chord that passes through the focus.

True — every chord through the focus is a "focal chord"; the sum of the two half-lengths from the focus is smallest when the chord is perpendicular to the axis. Tilt the chord and one end must slide up the arm to a point farther from the focus, so the total length only grows — meaning the perpendicular focal chord, the latus rectum (

4 a

), is the shortest.

True or false: If a parabola's equation contains

x^{2}

, it opens left or right.

False — it opens up or down. The axis of symmetry is the axis of the non-squared variable, so

x^{2}

means the

y

-axis is the axis and the curve opens vertically.

Spot the error

Spot the error: "For

y^{2} = 20 x

, the focus is

(20, 0)

The coefficient

20

equals

4 a

, so

a = 5

; the focus is

(5, 0)

, not

(20, 0)

. The mistake is treating the coefficient as

a

Spot the error: "For

y^{2} = 4 a x

, the directrix is

x = a

because

a

is the special number."

The directrix is on the opposite side of the vertex from the focus, so it is

x = - a

. The vertex

(0, 0)

is the midpoint between focus

(a, 0)

and directrix

x = - a

Spot the error: "

x^{2} = - 4 a y

opens to the left because of the minus sign."

The minus sets direction along the axis of the non-squared variable's opposite:

x^{2}

means vertical opening, and the minus makes it open downward, not leftward.

Spot the error: "The vertex of

y^{2} = 4 a x

is at

(a, 0)

That is the focus. The vertex is at the origin

(0, 0)

; it sits midway between focus

(a, 0)

and directrix

x = - a

Spot the error: "I set up the definition as

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

and solved."

The distance to the directrix is

∣ x + a ∣

, an absolute value, so before squaring you must know the right-hand side is non-negative — squaring is only a safe, reversible step when both sides are known to be $\geq 0$ . Writing

x + a

silently assumes

x + a \geq 0

. The clean fix is to square

∣ x + a ∣

directly, since

(∣ x + a ∣)^{2} = (x + a)^{2}

holds for every

x

regardless of sign; the left side

\dots

is already

\geq 0

, so both sides are non-negative and squaring introduces no false roots.

Spot the error: "A parabola with focus

(3, 0)

and directrix

x = 3

has vertex

(3, 0)

Here the directrix passes through the focus, so this is not a valid parabola at all. A directrix must not contain the focus.

Spot the error: "Length of latus rectum of

x^{2} = - 8 y

- 8

Length is always positive. The coefficient magnitude is

8 = 4 a

, so the latus rectum length is

8

; the minus only tells you it opens downward.

Spot the error: "The parabola

y^{2} = 4 a x

is a function

y = f (x)

For each

x > 0

there are two

y

-values (

\pm 4 a x

), so it fails the vertical-line test and is not a function of

x

. It is a function of

y

instead.

Why questions

Why is the coefficient in the standard form written as

4 a

rather than a single letter like

k

Because

4 a

is exactly the latus rectum length, so writing it this way lets you read off the width instantly and keeps

a

as the clean focus-to-vertex distance.

Why do we square both sides in the derivation instead of solving the square roots directly?

The two quantities being equated are distances — a square root and an absolute value — so each side is

\geq 0

automatically. Squaring is only reversible when both sides are known non-negative, and here that condition is met, so squaring introduces no false roots while removing the root and the bars in one stroke, leaving clean polynomials.

Why is the axis of symmetry perpendicular to the directrix rather than parallel to it?

The focus and directrix must be balanced by reflection; the only line that reflects the directrix onto itself and passes through the focus is the perpendicular through

F

, which is why that line is the axis.

Why do we insist

a > 0

and put the sign inside the equation instead of letting

a

be negative?

Keeping

a

a positive length means

a

4 a

, and the focus distance always mean "a genuine length"; direction is then read purely from the sign in front, avoiding sign confusion in the focus and directrix.

Why does a very small

a

produce a very "sharp" parabola near the vertex?

Small

a

pushes the focus close to the vertex and the directrix close on the other side; the equidistance condition then forces the curve to hug the axis tightly, giving a narrow, sharp bend.

Why does swapping

x \leftrightarrow y

in the derivation turn a horizontal parabola into a vertical one?

The whole derivation is symmetric in the roles of the two coordinates; swapping them swaps which axis is the axis of symmetry, so a

y^{2} = 4 a x

becomes

x^{2} = 4 a y

— a

9 0^{\circ}

role swap.

Edge cases

Edge case: Sign-inside vs make-

a

-negative — work

y^{2} = - 4 a x

from scratch with

a > 0

Take focus

(- a, 0)

and directrix

x = a

(mirror image of the right-opening set-up). The definition gives

(x + a)^{2} + y^{2} = ∣ x - a ∣

; squaring,

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

, and expanding cancels

x^{2} + a^{2}

to leave

y^{2} = - 4 a x

. Here

a

stayed positive (

a

is still the focus-to-vertex length); the minus in the equation is what encodes "opens left". Had we instead kept the form

y^{2} = 4 A x

and set

A = - a < 0

, we'd get the same curve but with a negative "

a

", which muddies the focus formula. Sign-inside keeps every length honest.

Edge case: What happens to the "parabola" as

a \to 0

The focus, vertex, and directrix all collapse toward the origin; the curve degenerates into the axis itself (a doubled straight line through the vertex) — no genuine parabola survives.

Edge case: Is a single point ever a valid parabola?

No — the definition needs a fixed point and a fixed line apart from each other. With no separation the locus is not a curve, so a lone point is a degenerate limit, not a parabola.

Edge case: Can the vertex of a standard-form parabola lie anywhere other than the origin?

Not for the standard forms

y^{2} = 4 a x

etc. — those are pinned with vertex at the origin. A shifted vertex needs the translated form

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

, which is beyond the standard four.

Edge case: On

y^{2} = 4 a x

, what is the point where

y = 0

, and is it special?

It is the vertex

(0, 0)

— the single point where the curve meets its own axis, the closest point to the focus, and the turning point of the two arms.

Edge case: For the point

(a, 2 a)

y^{2} = 4 a x

, why is its distance to the focus exactly

2 a

It is a latus-rectum endpoint, so its distance to the directrix

x = - a

a - (- a) = 2 a

; by the defining property its distance to the focus equals that,

2 a

— confirming the definition holds there.

Recall One-line self audit

If someone hands you $y^{2} = - 4 a x$ , can you name focus, directrix, axis, and opening without hesitating? → Focus $(- a, 0)$ , directrix $x = a$ , axis $y = 0$ , opens left. If any of those made you pause, re-read the parent section 4.

Connections

Conic Sections — Overview — where $e = 1$ sits among the conics
Eccentricity and focus-directrix definition — the ratio these traps rely on
Ellipse — standard forms and Hyperbola — standard forms — contrast the sign-and-shape logic
Tangents and Normals to a Parabola — next depth once these are solid
Coordinate Geometry — Distance Formula — the tool inside the "spot the error" derivation
Projectile Motion — where wide-vs-narrow $a$ intuition pays off