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Ellipse Equation

Ellipse: (x−h)²/a²+(y−k)²/b²=1, center (h,k), semi-axes a≥b. Eccentricity e=c/a where c²=a²−b². Foci at (±c,0) from center. Area A=πab. Circle is e=0.

Concept Fundamentals
(x−h)²/a²+(y−k)²/b²=1
Standard
e=c/a, c²=a²−b²
Eccentricity
A=πab
Area
±c from center
Foci

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Circle: a=b, e=0. Parabola: e=1 limit. Sum of distances from any point to foci = 2a. Planetary orbits are ellipses with Sun at one focus.

Key quantities
(x−h)²/a²+(y−k)²/b²=1
Standard
Key relation
e=c/a, c²=a²−b²
Eccentricity
Key relation
A=πab
Area
Key relation
±c from center
Foci
Key relation

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Why: Ellipses appear in orbits (Kepler), architecture (arches), and optics (reflectors). Eccentricity e: 0=circle, 1=parabola limit. Foci: sum of distances to foci is constant.

How: Standard form: (x−h)²/a²+(y−k)²/b²=1. a≥b. c=√(a²−b²). Foci at (h±c,k) for horizontal. Area=πab. Perimeter ≈ π[3(a+b)−√((3a+b)(a+3b))].

Circle: a=b, e=0. Parabola: e=1 limit.Sum of distances from any point to foci = 2a.
Sources:Khan Academy — EllipseWolfram MathWorld — Ellipse

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Ellipse EquationEnter center (h,k) and semi-axes a, b

Ellipse Parameters

Visualization

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🧮 Fascinating Math Facts

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(x−h)²/a²+(y−k)²/b²=1, e=c/a.

— Conic Sections

π

Area = πab; foci at ±c from center.

— Formulas

Key Takeaways

  • • An ellipse is the set of points where the sum of distances to two foci is constant.
  • • Standard form: frac(x−h)2a2+frac(y−k)2b2=1\\frac{(x-h)^2}{a^2}+\\frac{(y-k)^2}{b^2}=1 (horizontal) or with a,b swapped (vertical).
  • • Eccentricity e=c/ae=c/a is always between 0 and 1; e=0 gives a circle.
  • • Area = piab\\pi ab; perimeter has no closed form but Ramanujan's approximation is accurate.
  • • The foci lie along the major axis at distance c=sqrta2−b2c=\\sqrt{a^2-b^2} from the center.

Did You Know?

Planetary Orbits

Planetary orbits are ellipses with the Sun at one focus. Kepler's first law states this.

Whispering Galleries

In elliptical rooms, sound from one focus reflects to the other — used in St. Paul's Cathedral.

Ellipse vs Circle

A circle is a special ellipse where a = b, so eccentricity e = 0.

Oval vs Ellipse

An oval is any egg-like shape; an ellipse has a precise mathematical definition.

Elliptical Gears

Elliptical gears are used in some machinery for variable speed transmission.

Conic Sections

Ellipses, parabolas, and hyperbolas are all conic sections — slices of a cone.

Understanding Ellipse Equations

An ellipse has two axes: the major axis (longer) and minor axis (shorter). The semi-major aa and semi-minor bb define the shape.

frac(x−h)2a2+frac(y−k)2b2=1\\frac{(x-h)^2}{a^2}+\\frac{(y-k)^2}{b^2}=1

Horizontal ellipse: a is the semi-major (x-direction).

Expert Tips

Orientation

If a > b, the ellipse is horizontal (major axis along x). If b > a, it is vertical.

Foci Location

Foci are at (h±c, k) for horizontal, (h, k±c) for vertical, where c² = a² − b².

Perimeter

No simple formula exists. Ramanujan's approximation is commonly used.

General Form

Ax² + Cy² + Dx + Ey + F = 0. For an ellipse, A and C have the same sign.

Frequently Asked Questions

What is the standard form of an ellipse?

For a horizontal ellipse: (x-h)²/a² + (y-k)²/b² = 1, where (h,k) is the center, a is semi-major, b is semi-minor.

How do I find the foci?

Foci are at distance c = √(a² - b²) from the center along the major axis.

What is eccentricity?

e = c/a, where 0 ≤ e < 1. e = 0 for a circle; as e → 1 the ellipse becomes more elongated.

Can an ellipse have a = b?

Yes — that gives a circle. The foci coincide at the center.

What is the area of an ellipse?

Area = πab. This generalizes the circle formula πr² when a = b = r.

Is there an exact formula for ellipse perimeter?

No. It involves elliptic integrals. Ramanujan's approximation is accurate for most purposes.

How does orientation affect the equation?

For vertical ellipses, swap a and b in the denominators: the larger goes under (y-k)².

How to Use This Calculator

  1. Enter center (h, k) and semi-axes a, b. Choose orientation (horizontal or vertical).
  2. Click a sample example to auto-fill and see results.
  3. View the visualization with center, foci, and ellipse curve.
  4. Check the step-by-step solution for standard form and eccentricity.
  5. Copy results to share or paste into assignments.

Note: Semi-axes must be positive. Perimeter uses Ramanujan's approximation. Results are for educational use.

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