MathIsimple

Parabola

Learn the parabola through the focus–directrix lens, then upgrade to focal chord identities and line intersection techniques used in exams.

Advanced
Conic Sections
Focus + Directrix
Quick Facts
Short formulas that unlock many parabola questions.

Definition

Distance to focus equals distance to directrix

Standard forms

y² = ±2px, x² = ±2py (p>0)

Focus/directrix

For y²=2px: F(p/2,0), directrix x=−p/2

Latus rectum

Length = 2p

Focal chord

For y²=2px: x₁x₂=p²/4, y₁y₂=−p²

Parabola Calculator (Interactive)
Choose a direction, enter p, and instantly get the equation, focus, directrix, and latus rectum with SVG preview.

Select direction, enter p, then click Calculate.

Core Knowledge

1) Definition & Standard Forms
Equidistance definition and the four standard equations.

Focus–directrix definition

A parabola is the set of points P such that:

dist⁡(P,focus)=dist⁡(P,directrix)\operatorname{dist}(P,\text{focus})=\operatorname{dist}(P,\text{directrix})

This definition is your safest starting point when a problem mentions a directrix or distance equality.

Four standard forms (p > 0)

y2=2px,y2=−2px,x2=2py,x2=−2pyy^2 = 2px,\quad y^2=-2px,\quad x^2=2py,\quad x^2=-2py

Sign controls the opening direction; swapping x and y rotates the parabola.

One form to remember

Memorize the “right-opening” form:

y2=2pxy^2 = 2px

Then generate the other three by symmetry (sign flips) and rotation (swap x and y).

Focus–directrix definition (right-opening)

xydirectrixx = −p/2F(p/2,0)

A point P lies on the parabola iff its distance to the focus equals its distance to the directrix.

Four standard directions

xyRight: y² = 2pxLeft: y² = −2pxUp: x² = 2pyDown: x² = −2py

The sign determines the opening direction; swapping x and y rotates the parabola.

2) Geometric Properties
Focus, directrix, focal radius, latus rectum, and optics.

Focus & directrix (right-opening)

y2=2px(p>0)y^2=2px\quad (p>0)Focus F(p2,0),Directrix x=−p2\text{Focus }F\left(\frac{p}{2},0\right),\qquad \text{Directrix }x=-\frac{p}{2}

Other directions follow by symmetry/rotation.

Focal radius

For y² = 2px and P(x₀,y₀) on the parabola, a standard identity is:

PF=x0+p2PF = x_0 + \frac{p}{2}

It comes directly from the focus–directrix definition because dist(P, directrix) = x₀ + p/2.

Latus rectum (length = 2p)

xyFlength = 2p

For y² = 2px, the latus rectum endpoints are (p/2, ±p).

Optical property

Parabolas focus parallel rays: a ray parallel to the axis reflects through the focus. This is why satellite dishes are paraboloids.

Exam connection: you may translate “parallel” into slope constraints and use tangency/normal line facts.

3) Parabola with a Line
Eliminate → quadratic → Δ/Vieta; symmetry shortcuts.

Elimination + discriminant

Combine a line with a parabola to obtain a quadratic in one variable. Tangency corresponds to:

Δ=0\Delta = 0

Vieta principle

For intersection points A,B, use Vieta to compute x₁+x₂, x₁x₂ (or y₁+y₂, y₁y₂). Many area/length invariants simplify dramatically.

Symmetry

The axis of symmetry is the line passing through the focus and perpendicular to the directrix. Many “midpoint on axis” facts come from symmetry.

Chord length and slope

When the line is written as x = my + n (for y²=2px), the algebra often becomes cleaner because y is the “squared” variable.

A chord length template (exam-style)

If a line intersects the parabola at A and B, then AB is often computed via a Vieta shortcut: first find the intersection roots, then convert the coordinate difference into a length using the line direction.

Key idea: avoid solving A and B explicitly when the problem only asks for AB, midpoint, or area.

4) Focal Chords (High-Yield Identity)
Two identities that show up repeatedly in advanced problems.

For y² = 2px

If a chord through the focus intersects the parabola at points A(x₁,y₁) and B(x₂,y₂), then:

x1x2=p24,y1y2=−p2x_1x_2=\frac{p^2}{4},\qquad y_1y_2=-p^2

Why it matters

These identities let you replace “complicated coordinates” with constants. They power length/area/min-max problems.

Quick derivation hint

Use the parameter form x=(p/2)t², y=pt and the focal chord relation t₁t₂ = −1.

Chord length (advanced)

Once you parameterize points, chord length and angle constraints often become algebraic inequalities in t.

5) Extensions (Optics & Applications)
Real-world meaning of focus + directrix geometry.

Parabolic reflectors

Parabolic mirrors turn parallel rays into a focal point and vice versa. This is the geometry behind telescopes, headlights, and satellite dishes.

Quadratic functions

In algebra, a parabola is the graph of a quadratic function. In analytic geometry, the focus–directrix definition gives extra structure for distance problems.

6) Common Pitfalls
Stop losing points to sign and orientation errors.

1) Mixing 2p and 4p conventions

Some books use y² = 4ax. On this page we use y² = 2px. Always convert carefully.

2) Wrong opening direction

The sign in the equation controls direction. If your focus ends up on the wrong side of the vertex, re-check the sign.

3) Forgetting Δ for tangency

A tangent line usually appears via elimination and Δ = 0, not by guessing a point.

Practice Quiz

Practice Quiz
10
Questions
0
Correct
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Accuracy
1
A parabola is the set of points equidistant from a focus and a:
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2
Which is a standard form of a parabola opening to the right?
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3
For y² = 2px (p>0), the focus is located at:
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4
For y² = 2px (p>0), the directrix is:
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5
The latus rectum length of y² = 2px equals:
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6
A parabola has eccentricity:
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7
For y² = 2px, if a point P(x0,y0) lies on the parabola, a common focal radius expression is:
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8
For y² = 2px, focal chord endpoints (x1,y1),(x2,y2) satisfy:
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9
A key test for line–parabola tangency after substitution is:
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10
Which optical statement is correct for a parabola?
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