Parabola
A parabola is a conic section formed by the intersection of a right circular cone with a plane parallel to one of the cone's slant sides. In coordinate geometry, a parabola is defined as the set of all points in a plane that are equidistant from a fixed point called the focus and a fixed line called the directrix. This elegant distance property gives the parabola its characteristic U-shaped curve and underlies every formula and measurement associated with it. Parabolas appear throughout mathematics, physics, and engineering. The path of a projectile under gravity traces a parabolic arc, satellite dishes and car headlight reflectors are paraboloid surfaces (three-dimensional versions of parabolas), and the cables of a suspension bridge form a parabolic shape under uniform load. Understanding the standard equation of a parabola — and key features like its vertex, axis of symmetry, focus, directrix, and latus rectum — is therefore both theoretically important and practically powerful. In this article we focus on parabolas whose axis of symmetry is parallel to a coordinate axis. There are two orientations: vertical (opening up or down) and horizontal (opening right or left). The parameter p, which equals the distance from the vertex to the focus (and also from the vertex to the directrix), completely determines the shape of the parabola. The latus rectum is a chord through the focus perpendicular to the axis; its length 4|p| gives an immediate visual measure of how wide the parabola is at the focus.
- Defining a Parabola via the Focus-Directrix Property
- Deriving the Vertical Parabola Equation
- Horizontal Parabola and Shifted Vertex
- Locating the Focus, Directrix, and Axis
- The Latus Rectum
Defining a Parabola via the Focus-Directrix Property
Let the vertex be at the origin V = (0, 0). Place the focus at F = (0, p) directly above the vertex (we will handle other orientations by analogy). The directrix is the horizontal line y = -p, at the same distance |p| below the vertex. For any point P = (x, y) on the parabola, the definition requires |PF| = |Pd|, where |Pd| is the perpendicular distance from P to the directrix.
Deriving the Vertical Parabola Equation
Applying the distance formula: √(x² + (y - p)²) = |y + p|. Squaring both sides: x² + (y - p)² = (y + p)². Expanding: x² + y² - 2py + p² = y² + 2py + p². Simplifying: x² = 4py. This is the standard form for a vertical parabola with vertex at the origin. If p > 0 the parabola opens upward; if p < 0 it opens downward.
Horizontal Parabola and Shifted Vertex
By symmetry, swapping the roles of x and y gives a horizontal parabola: y² = 4px. Here the focus is at (p, 0) and the directrix is x = -p. For a parabola with vertex at (h, k), the standard forms become (x - h)² = 4p(y - k) for a vertical axis and (y - k)² = 4p(x - h) for a horizontal axis. These four cases cover all axis-aligned parabolas.
Locating the Focus, Directrix, and Axis
From (x - h)² = 4p(y - k): vertex = (h, k), focus = (h, k + p), directrix: y = k - p, axis of symmetry: x = h. From (y - k)² = 4p(x - h): vertex = (h, k), focus = (h + p, k), directrix: x = h - p, axis of symmetry: y = k. In both cases |p| is the focal length.
The Latus Rectum
The latus rectum (Latin: 'right side') is the chord of the parabola that passes through the focus and is perpendicular to the axis of symmetry. To find its length for x² = 4py, substitute y = p (the y-coordinate of the focus) into the equation: x² = 4p · p = 4p², so x = ±2p. The two endpoints of the latus rectum are (-2p, p) and (2p, p), giving a length of |2p - (-2p)| = 4|p|. The latus rectum length is always 4|p|, regardless of orientation, and is a useful measure of the parabola's 'width' at the focus.
The Key Equation
- h — x-coordinate of vertex (none): Horizontal position of the vertex; shifts the parabola left or right.
- k — y-coordinate of vertex (none): Vertical position of the vertex; shifts the parabola up or down.
- p — focal parameter (focal length) (none): Signed distance from vertex to focus. Positive p opens upward; negative p opens downward. The focus is at (h, k+p) and the directrix is y = k-p.
- 4p — latus rectum length coefficient (none): Four times p; the full length of the latus rectum is |4p|, measuring the width of the parabola at the focus.
Worked Example
Problem. A parabola has its vertex at (2, -3) and its focus at (2, 1). Find: (a) the standard equation of the parabola, (b) the equation of the directrix, (c) the length and endpoints of the latus rectum.
Answer. Equation: (x−2)² = 16(y+3); Directrix: y = −7; Latus rectum length: 16, with endpoints (−6, 1) and (10, 1).
Origins · 1 minApollonius of Perga and how this was discovered
Apollonius wrote the monumental eight-volume treatise Conics (Κωνικά) around 200 BCE, in which he systematically studied ellipses, parabolas, and hyperbolas as sections of a cone. He coined the very names 'parabola,' 'ellipse,' and 'hyperbola' based on the relationship between the ordinate squared and the abscissa — terminology still used today. His work on the focus-directrix definition and the optical reflective property of parabolas laid the foundation for all later analytic treatments. The word 'parabola' comes from the Greek parabole (παραβολή) meaning 'a placing side by side' or 'comparison,' referring to the exact equality in the focus-directrix definition — the distance to the focus equals the distance to the directrix, no more and no less.
Related Concepts
- Ellipse: Standard Equation and Foci — An ellipse is another conic section formed by a plane cutting a cone at an angle. Comparing the standard forms of the ellipse and parabola deepens understanding of how the eccentricity e determines the type of conic: e = 1 gives a parabola, e < 1 gives an ellipse.
- Hyperbola: Standard Equation and Asymptotes — The hyperbola (eccentricity e > 1) completes the family of conic sections alongside the parabola and ellipse. All three share the focus-directrix definition with different eccentricities.
- Distance Formula — The distance formula is the core algebraic tool used to derive the standard equation of a parabola from its focus-directrix definition, by equating distances from a general point to the focus and to the directrix.
- Quadratic Functions and Vertex Form — The standard parabola equation (x-h)² = 4p(y-k) can be rewritten as y = (1/4p)(x-h)² + k, which is the vertex form y = a(x-h)² + k of a quadratic function with a = 1/(4p). Understanding both forms links conic sections to algebra.
- Completing the Square — Converting a general second-degree equation like x² + Dx + Ey + F = 0 into standard parabola form requires completing the square — an essential algebraic technique in coordinate geometry.