Ellipse

An ellipse is a closed, oval-shaped curve that is one of the four classical conic sections — curves formed by slicing a double cone with a plane. Specifically, an ellipse results when the plane cuts through one nappe of the cone at an angle that is less steep than the cone's side, but not perpendicular to the axis. Every ellipse has two special interior points called foci (singular: focus), and the defining geometric property is that for any point P on the ellipse, the sum of its distances to the two foci is constant: PF₁ + PF₂ = 2a, where 2a is the length of the longest diameter. Ellipses appear throughout mathematics, science, and engineering. Planets orbit the Sun in elliptical paths — a fact discovered by Johannes Kepler in the early 17th century. Elliptical mirrors and rooms have the remarkable property that sound or light emitted from one focus is reflected perfectly to the other focus, which is why certain architectural spaces are called 'whispering galleries.' Understanding the algebraic equation of an ellipse, its key measurements (semi-major axis, semi-minor axis, foci), and its eccentricity gives you powerful tools to analyse these shapes precisely. In coordinate geometry, the standard equation of an ellipse centred at the origin depends on whether the major axis is horizontal or vertical. The eccentricity e (a number between 0 and 1) measures how 'stretched' the ellipse is: when e = 0 the shape is a perfect circle, and as e approaches 1 the ellipse becomes increasingly elongated. The latus rectum is a chord through a focus perpendicular to the major axis, and its length provides a compact way to describe the ellipse's width at the focus.

Key takeaways
  • The Focal Definition
  • Deriving the Standard Equation
  • Horizontal vs. Vertical Major Axis
  • Eccentricity
  • Latus Rectum
  • Shifting the Centre

The Focal Definition

Place two foci F₁(−c, 0) and F₂(c, 0) on the x-axis. A point P(x, y) lies on the ellipse if PF₁ + PF₂ = 2a, where a > c > 0. The constant 2a is the total path length; it must exceed 2c (the distance between the foci) for a non-degenerate ellipse.

Deriving the Standard Equation

Starting from √[(x+c)²+y²] + √[(x−c)²+y²] = 2a, isolate one radical and square twice. After simplification you obtain x²/a² + y²/b² = 1, where b² = a² − c². Here a is the semi-major axis (longest half-width), b is the semi-minor axis (shortest half-width), and c is the focal distance from the centre.

Horizontal vs. Vertical Major Axis

If a² is under x², the major axis is horizontal and the ellipse is wider than it is tall; vertices are at (±a, 0) and co-vertices at (0, ±b). If a² is under y², the major axis is vertical; vertices are at (0, ±a) and co-vertices at (±b, 0). The foci always lie along the major axis at distance c = √(a²−b²) from the centre.

Eccentricity

The eccentricity is defined as e = c/a, and always satisfies 0 ≤ e < 1. When e = 0, c = 0 so both foci coincide at the centre and the ellipse is a circle. As e → 1, the ellipse becomes very elongated. For example, Earth's orbit has e ≈ 0.017 (nearly circular), while Halley's Comet has e ≈ 0.967 (highly elongated).

Latus Rectum

The latus rectum is the chord of the ellipse passing through a focus and perpendicular to the major axis. To find its length, substitute x = c into x²/a² + y²/b² = 1: y² = b²(1 − c²/a²) = b²·b²/a² = b⁴/a², so y = ±b²/a. The full length of the latus rectum is therefore 2b²/a. This value equals 2a when e = 0 (circle) and approaches 0 as e → 1.

Shifting the Centre

For an ellipse centred at (h, k) instead of the origin, replace x with (x−h) and y with (y−k): (x−h)²/a² + (y−k)²/b² = 1. All properties — axes lengths, eccentricity, latus rectum — remain the same; only the positions of the centre, foci, and vertices shift by (h, k).

The Key Equation

x2a2+y2b2=1,b2=a2−c2,e=ca,ℓ=2b2a\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, \quad b^2 = a^2 - c^2, \quad e = \frac{c}{a}, \quad \ell = \frac{2b^2}{a}
  • a — Semi-major axis (length units): Half the length of the longest diameter; a > b > 0. Vertices lie at (±a, 0).
  • b — Semi-minor axis (length units): Half the length of the shortest diameter; co-vertices lie at (0, ±b).
  • c — Focal distance (length units): Distance from the centre to each focus; c = √(a²−b²).
  • e — Eccentricity (dimensionless): Ratio e = c/a; measures elongation. For an ellipse, 0 ≤ e < 1.
  • \ell — Latus rectum (length units): Length of the chord through a focus perpendicular to the major axis; ℓ = 2b²/a.

Worked Example

Problem. An ellipse has equation 9x² + 25y² = 225. Find: (a) the semi-axes a and b, (b) the foci, (c) the eccentricity, and (d) the length of the latus rectum.

a=5,  b=3,  F1,2=(±4,0),  e=45,  ℓ=185a=5,\; b=3,\; F_{1,2}=(\pm 4,0),\; e=\tfrac{4}{5},\; \ell=\tfrac{18}{5}

Answer. a = 5, b = 3, foci at (±4, 0), eccentricity e = 4/5, latus rectum = 18/5 units.

Figure
Diagram
xyabF₁(−c,0)F₂(c,0)cLatusrectum2b²/a(a,0)(−a,0)(0,b)(0,−b)OPPF₁PF₂Ellipse: x²/a² + y²/b² = 1PF₁ + PF₂ = 2a (constant)
Origins · 1 minApollonius of Perga and how this was discovered
AO
Apollonius of Perga
-262–-190 · Greek

Apollonius wrote the landmark eight-volume treatise Conics (Κωνικά) around 200 BCE, in which he systematically studied and named the conic sections — ellipse, parabola, and hyperbola — deriving their properties from a single double cone. He introduced the term 'ellipse' from the Greek word meaning 'falling short,' reflecting the geometric relationship between the shape and its defining cone. His rigorous geometric approach to conics remained the definitive reference for nearly 1800 years. Apollonius was called 'The Great Geometer' by his contemporaries, a title that has endured for over two millennia. He also made significant contributions to astronomy, developing early epicycle models that would influence Ptolemy's geocentric system.

  • Parabola — Another conic section, formed when the plane is parallel to a slant edge of the cone; its equation and eccentricity (e = 1) contrast directly with the ellipse (e < 1).
  • Hyperbola — The remaining open conic section (e > 1); shares the same standard-form structure x²/a² − y²/b² = 1 and the concept of foci, making comparison with the ellipse illuminating.
  • Circle — A special case of the ellipse where a = b and eccentricity e = 0; understanding this degenerate case reinforces the ellipse's defining parameters.
  • Distance Formula — The algebraic derivation of the ellipse's standard equation relies entirely on applying the distance formula to the two focal radii.
  • Focus–Directrix Property — All conics can be defined using a focus and a directrix line; for an ellipse e = PF/PD < 1, unifying the definitions of all conic sections.