Parabola
The set of points equidistant from a focus and a directrix — the shape of projectile paths and satellite dishes.
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Formulas and key values
- Cartesian equation
- Focus
- Directrix
- Latus rectumIn the graph = 4
- Eccentricity
p is the distance from the vertex to the focus.
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Start the video01 · Definition
What is the Parabola?
The parabola is the locus of points equidistant from a fixed point (the focus) and a fixed line (the directrix). It is also the conic section cut by a plane parallel to the cone’s slant side. Menaechmus studied it around 350 BC while attacking the cube-doubling problem; Apollonius gave it its name.
Two properties make it ubiquitous. Galileo showed that projectiles under uniform gravity follow parabolas. And its reflective property — rays from the focus emerge parallel to the axis, and incoming parallel rays converge to the focus — puts a parabola inside every satellite dish, car headlight, and radio telescope.
02 · Properties
Key properties
- Focus at (0, p) and directrix y = −p for x² = 4py.
- All parabolas are geometrically similar — they differ only by scale.
- Reflective property: focal rays exit parallel to the axis.
- The latus rectum (chord through the focus, parallel to the directrix) has length 4p.
- Eccentricity is exactly 1 — the boundary between ellipses and hyperbolas.
03 · Questions
Parabola: common questions
- What is the equation of the Parabola?
- Cartesian equation: x^2 = 4py, y = (x^2)/4p. p is the distance from the vertex to the focus.
04 · Labs
Interactive labs
Derivative as the slope of a tangent
Why is the derivative the slope of the tangent line?
Secant slope · Limit definition · Tangent line
Open labVertex form: how a, h and k move a parabola
What do a, h and k each do to y = a(x − h)² + k?
Vertex form · Translations · Axis of symmetry
Open labConic sections and eccentricity
How can one number turn a circle into an ellipse, a parabola and a hyperbola?
Focus and directrix · Eccentricity · Ellipse, parabola, hyperbola
Open labProjectile motion: two motions at once
Why does a thrown ball follow a parabola, and which angle throws farthest?
Independent components · Horizontal launch · Range and angle
Open labThe parabola: equal distances to a focus and a directrix
Why is the focal radius of y² = 2px just x + p/2, and how long is a chord through the focus?
Focus and directrix · Standard equation y² = 2px · Focal radius
Open lab05 · Wonders
See it in a math wonder
06 · Keep exploring