Focus-Directrix Parabola Width
A parabola's width at the focus is exactly twice its focus–directrix distance, so one measured distance reveals the scale of the whole curve.
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If you know only the distance between a parabola's focus and directrix, can you predict how wide it opens?
- focus-directrix-definition
- A parabola is the set of points equidistant from a fixed point (focus) and a fixed line (directrix).
- focal-distance-d
- The perpendicular distance between the focus and the directrix, often denoted d, sets the scale of the parabola.
- latus-rectum
- The chord through the focus perpendicular to the axis; its length is the parabola's width at the focus.
- width-relation
- For any parabola, the latus rectum length equals twice the focus–directrix distance: width = 2d.
The width at the focus is equal to the focus–directrix distance.
Show by coordinate geometry that the latus rectum length is 2d, so it is twice the focus–directrix distance, not equal to it.
Changing the focus–directrix distance changes only where the parabola sits, not how wide it is.
Demonstrate that the focus–directrix distance controls the scale and therefore the width of the parabola.
- Distance formula
- Definition of a parabola as the set of points equidistant from a focus and a directrix
- Basic coordinate graphing
- General conic section equations for ellipse and hyperbola
- Rotated or tilted parabolas
- Applications in optics and projectile motion
- Deriving vertex form from arbitrary points
- Given a focus–directrix distance, compute the width at the focus.
- Given focus and directrix coordinates, find d and the width at the focus.
- Explain why the latus rectum is twice the focus–directrix distance.
- Given a new parabola described only by its focus and directrix, predict its width at the focus before graphing it.
High school geometry or introductory precalculus learners who can use the distance formula and are comfortable graphing simple equations.
- 01Focus, Directrix, and the Width QuestionslideOrientationObserve
Introduce the focus–directrix definition and frame the key question: can one distance tell us the parabola's width?
- A parabola is all points equidistant from a focus and a directrix
- The distance between focus and directrix is one clean number
- We want the width of the parabola at the focus
- 02Make a PredictionquizPredictionPredict
Ask learners to predict how focus–directrix distance controls width before seeing the derivation.
- Choose the relationship you expect
- Committing to a guess makes the proof more memorable
- 03Explore the WidthinteractiveModel buildingConstruct
Drag the focus and directrix and watch the parabola and its width at the focus update in real time.
- Move the focus or directrix to change d
- Notice the latus rectum width
- Try doubling d and compare the width
- 04Why the Width Is Twice dslideMisconception repairObserve
Show a short coordinate proof that the latus rectum length equals 2d, not d.
- Place focus at (p,0) and directrix x = -p, so d = 2p
- An endpoint at the focus has horizontal distance d to the directrix
- By equidistance, its y-coordinate is ±d, so full width is 2d
- 05Check the RelationshipquizPracticeApply
Practice converting a focus–directrix distance into a width at the focus.
- A parabola has d = 4. How wide is it at the focus? (Answer: 8)
- A parabola has width 10 at the focus. What is d? (Answer: 5)
- 06Apply It to a Real ParabolaquizApplicationApply
Use focus and directrix coordinates to find the focus–directrix distance and then the width.
- Find d from focus (2,0) and directrix x = -1
- Width = 2d
- The endpoints at the focus are at y = ±d
- 07The Big Idea: One Distance Sets the ScaleslideSynthesisExplain
Synthesize the relationship and show how it gives a fast way to sketch parabolas.
- Width at focus = 2 × focus–directrix distance
- Focal distance d controls scale, not position
- Use d to place the endpoints before drawing the curve
- 08Check Your UnderstandingquizAssessmentApply
Assess the full relationship: definitions, d, latus rectum, and applying width = 2d to a fresh example.
- Identify the focus and directrix in a standard setup
- Compute width from d
- Explain why width ≠ d
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