Ellipse - Equation, Formula, Properties, Graphing

What Is an Ellipse?

An ellipse is the locus of all points in a plane whose distances to two fixed points — the foci — sum to a constant value. Mathematically, for any point $P$ on the ellipse:

$$|PF_1| + |PF_2| = 2a$$

where $F_1$ and $F_2$ are the foci and $2a$ is the length of the major axis.

An ellipse is also a conic section — formed by slicing a cone with a plane at a slight angle (not parallel to the base, not parallel to the slant side). A circle is the special case of an ellipse where both foci coincide at the centre.

The standard equation of an ellipse centred at the origin with major axis along the x-axis:

$$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, \quad a > b > 0$$

where $a$ is the semi-major axis length and $b$ is the semi-minor axis length.

What Are the Parts of an Ellipse?

What Is the Equation of an Ellipse?

Standard Form — Centred at Origin

For an ellipse centred at $(0, 0)$ with major axis along the x-axis:

$$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, \quad a > b$$

For major axis along the y-axis (taller than wide):

$$\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1, \quad a > b$$

Shifted Form — Centred at $(h, k)$

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

Relationship Between $a$, $b$, and $c$

The distance from the centre to each focus is $c$, where:

$$c = \sqrt{a^2 - b^2}$$

So the foci sit at $(\pm c, 0)$ for a horizontal ellipse, or $(0, \pm c)$ for a vertical ellipse.

How Do You Derive the Ellipse Equation?

The standard form $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ falls straight out of the focus-pair definition + the distance formula. Here's the derivation.

Setup. Place the centre at the origin, foci at $F_1 = (-c, 0)$ and $F_2 = (c, 0)$, and let $P = (x, y)$ be any point on the ellipse. By definition, the sum of distances to the foci is $2a$:

$$\sqrt{(x + c)^2 + y^2} + \sqrt{(x - c)^2 + y^2} = 2a$$

Isolate one radical.

$$\sqrt{(x + c)^2 + y^2} = 2a - \sqrt{(x - c)^2 + y^2}$$

Square both sides.

$$(x + c)^2 + y^2 = 4a^2 - 4a\sqrt{(x - c)^2 + y^2} + (x - c)^2 + y^2$$

Expand and cancel the $(x \pm c)^2 + y^2$ terms (after expanding, the $x^2, c^2, y^2$ cancel):

$$4cx = 4a^2 - 4a\sqrt{(x - c)^2 + y^2}$$

Solve for the remaining radical.

$$a\sqrt{(x - c)^2 + y^2} = a^2 - cx$$

Square again.

$$a^2\bigl[(x - c)^2 + y^2\bigr] = a^4 - 2a^2 cx + c^2 x^2$$

Expand and group.

$$a^2 x^2 - 2a^2 cx + a^2 c^2 + a^2 y^2 = a^4 - 2a^2 cx + c^2 x^2$$

$$\bigl(a^2 - c^2\bigr)x^2 + a^2 y^2 = a^2\bigl(a^2 - c^2\bigr)$$

Substitute $b^2 = a^2 - c^2$ (this is the definition of $b$ for an ellipse — it makes the focal-distance formula come out clean):

$$b^2 x^2 + a^2 y^2 = a^2 b^2$$

Divide by $a^2 b^2$:

$$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$

That's the standard ellipse equation — and the substitution $b^2 = a^2 - c^2$ (equivalently $c = \sqrt{a^2 - b^2}$) drops out as a consequence of the derivation, not a definition you have to memorise.

What Are the Properties of an Ellipse?

Beyond the parts list, an ellipse has six geometric properties that come up in problems and proofs.

1. Two axes of symmetry. The major and minor axes are both lines of symmetry. (A circle has infinite; a parabola has one.)

2. Two foci, constant distance-sum. For every point $P$ on the curve, $|PF_1| + |PF_2| = 2a$. This is the defining property and is often used as a problem-solving constraint.

3. Closed, bounded curve. Unlike parabolas and hyperbolas, an ellipse is a finite closed curve. It encloses a finite area.

4. Area = $\pi a b$. Generalises the circle's $\pi r^2$. (When $a = b = r$, this gives $\pi r^2$ ✓.)

5. Perimeter — no closed-form formula. Unlike area, the ellipse perimeter doesn't have a clean elementary expression. Ramanujan's approximation gives $P \approx \pi!\left[3(a + b) - \sqrt{(3a + b)(a + 3b)}\right]$, accurate to a few parts per million for moderate eccentricities.

6. Reflective property. Any ray emitted from one focus reflects off the ellipse and passes through the other focus. This is the principle behind whispering galleries and lithotripsy — a sound or shock wave from one focus arrives at the other focus all at the same instant.

How Do You Draw an Ellipse? (The Gardener's Method)

The geometric definition gives a beautifully simple physical construction — sometimes called the gardener's method or string-and-pins method — used since antiquity to mark elliptical flowerbeds and arenas.

Materials. Two pins (or pegs, or thumbtacks), a piece of string of length $2a$, and a pencil.

Steps.

1. Decide the major-axis length $2a$ and the focal distance $2c$ (with $c < a$).

2. Press the two pins into the drawing surface, $2c$ apart — those are the foci $F_1$ and $F_2$.

3. Tie the string into a loop of total length $2a + 2c$ (so when stretched around both pins, the loose part forms a triangle).

4. Place the pencil inside the loop, pull the string taut, and trace a complete circuit.

The pencil traces an ellipse because the string keeps the sum $|PF_1| + |PF_2| = 2a$ constant at every point. That's the focus-pair definition, made tangible.

Algebraic alternative — graphing from the equation. For $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$, the 6-step process below (Centre → Orientation → Vertices → Co-vertices → Foci → Curve) works the same way and is what you'll do on graph paper.

What Is Eccentricity?

The eccentricity $e$ of an ellipse measures how elongated it is — how much it deviates from being a perfect circle.

$$e = \frac{c}{a} = \sqrt{1 - \frac{b^2}{a^2}}$$

• $e = 0$ → perfect circle (foci coincide at centre)

• $0 < e < 1$ → ellipse (the closer to 1, the more stretched)

• $e = 1$ → parabola (degenerate case)

• $e > 1$ → hyperbola (the other side of conic sections)

Examples. Earth's orbit has eccentricity $e \approx 0.0167$ — almost circular. Halley's Comet has $e \approx 0.967$ — extremely elongated.

How Do You Graph an Ellipse?

For an ellipse $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$:

Step 1. Identify the centre $(h, k)$.

Step 2. Determine orientation. If $a > b$ under $x$, major axis is horizontal. If $a > b$ under $y$, major axis is vertical.

Step 3. Mark the vertices at distance $a$ from the centre along the major axis.

Step 4. Mark the co-vertices at distance $b$ from the centre along the minor axis.

Step 5. Compute $c = \sqrt{a^2 - b^2}$ and mark the foci on the major axis.

Step 6. Connect the four key points with a smooth oval curve.

Worked example. Graph $\frac{x^2}{25} + \frac{y^2}{9} = 1$.

• Centre: $(0, 0)$.

• $a^2 = 25$, $b^2 = 9$, so $a = 5$, $b = 3$.

• Major axis horizontal (under $x$). Vertices: $(\pm 5, 0)$. Co-vertices: $(0, \pm 3)$.

• $c = \sqrt{25 - 9} = 4$. Foci: $(\pm 4, 0)$.

• Eccentricity: $e = 4/5 = 0.8$.

Why Are Ellipses Important? (The Real-World GROUND)

"The orbit of every planet is an ellipse with the Sun at one of the two foci." — Johannes Kepler, Astronomia Nova, 1609.

The ellipse was one of the most important geometric discoveries in the history of science — not for its mathematics directly, but for what it revealed about the universe.

In 1609, the German astronomer Johannes Kepler published Astronomia Nova, in which he proved — using Tycho Brahe's data on Mars — that the orbit of every planet is an ellipse, not a circle, with the Sun at one focus. This was Kepler's First Law, and it overturned 2,000 years of Greek astronomy (which assumed circular orbits since Aristotle).

Newton later proved (1687) that elliptical orbits are a mathematical consequence of the inverse-square law of gravity — connecting geometry, physics, and the cosmos in one formula.

Real-world applications of the ellipse:

• Planetary orbits. Mercury, Venus, Earth, Mars, Jupiter — all elliptical. The Hubble Space Telescope and the International Space Station both orbit Earth in elliptical paths.

• Halley's Comet. Returns every 76 years on a highly elliptical orbit ($e \approx 0.967$).

• Whispering galleries. St. Paul's Cathedral in London and the U.S. Capitol Rotunda have elliptical domes — a whisper at one focus can be heard clearly at the other focus, because sound waves reflect off the ellipse from one focus to the other.

• Lithotripsy. Medical machines use elliptical reflectors to focus shock waves on kidney stones — the shock generator sits at one focus, the stone at the other.

• Architecture. The Colosseum in Rome is an ellipse (188m × 156m). Many oval running tracks are elliptical.

• Engineering — gears. Elliptical gears produce non-uniform rotation useful in some specialised machinery.

The ellipse was first studied by Apollonius of Perga in his Conics around 200 BCE — he gave it the name ellipsis (Greek for "falling short") and proved its main geometric properties.

Worked Example of Ellipse

Find the foci of the ellipse $\frac{x^2}{25} + \frac{y^2}{16} = 1$.

The intuitive (wrong) approach. A student in a hurry computes $c = \sqrt{a^2 + b^2}$ instead of $\sqrt{a^2 - b^2}$:

$$c \stackrel{?}{=} \sqrt{25 + 16} = \sqrt{41} \approx 6.4$$

Why it fails. The formula for the focal distance in an ellipse is $c = \sqrt{a^2 - b^2}$ (minus, not plus). The plus formula applies to hyperbolas, not ellipses.

The correct method.

$$c = \sqrt{a^2 - b^2} = \sqrt{25 - 16} = \sqrt{9} = 3$$

The foci are at $(\pm 3, 0)$.

Check. Eccentricity $e = c/a = 3/5 = 0.6$, which is between 0 and 1 ✓ (the ellipse range). The wrong answer of $c = 6.4 > a = 5$ would have produced $e > 1$, which is the hyperbola range — a contradiction.

At Bhanzu, our trainers walk through this wrong-path-first sequence intentionally — confusing the ellipse focal formula with the hyperbola version is one of the most common student archetypes. Once the student feels the sign-flip cost, the rule sticks.

What Are the Most Common Mistakes With Ellipses?

Mistake 1: Using $c = \sqrt{a^2 + b^2}$ instead of $c = \sqrt{a^2 - b^2}$

Where it slips in: Confusing the ellipse focal-distance formula with the hyperbola one.

Don't do this: $c = \sqrt{a^2 + b^2}$ for an ellipse.

The correct way: For ellipse, $c = \sqrt{a^2 - b^2}$ (minus). For hyperbola, $c = \sqrt{a^2 + b^2}$ (plus). The rusher who pattern-matches "Pythagorean-style formula" hits this.

Mistake 2: Misidentifying major vs minor axis

Where it slips in: Assuming the $x$-direction is always the major axis.

Don't do this: For $\frac{x^2}{9} + \frac{y^2}{25} = 1$, calling the major axis horizontal.

The correct way: The larger denominator is under the major axis variable. Here, $25 > 9$ is under $y$ — so the major axis is vertical, and $a^2 = 25$, $b^2 = 9$. The memorizer who pattern-matches "a is always under x" without checking which is larger hits this.

Mistake 3: Forgetting an ellipse can be a circle when $a = b$

Where it slips in: When the two denominators are equal, students sometimes don't recognise the resulting circle.

Don't do this: Treating $\frac{x^2}{16} + \frac{y^2}{16} = 1$ as a normal ellipse with computed $c$.

The correct way: When $a = b$, the ellipse becomes a circle of radius $a$. Eccentricity is 0, foci coincide at the centre. $\frac{x^2}{16} + \frac{y^2}{16} = 1$ simplifies to $x^2 + y^2 = 16$ — a circle of radius 4.

The Mathematicians Who Shaped Ellipse Theory

Apollonius of Perga (c. 240–c. 190 BCE, Greece) — Wrote Conics, the foundational treatise on parabolas, ellipses, and hyperbolas. Coined the term ellipse (Greek elleipsis, "falling short") and proved its main geometric properties two millennia before algebraic notation existed.

Johannes Kepler (1571–1630, Germany) — Proved in 1609 that planetary orbits are ellipses with the Sun at one focus (Kepler's First Law). This overturned 2,000 years of assumption that orbits must be circular, and gave Newton the geometric foundation for the inverse-square law of gravity.

Tycho Brahe (1546–1601, Denmark) — Made the most precise pre-telescope astronomical observations of his era, especially of Mars. Brahe's Mars data was what Kepler used to discover elliptical orbits — the empirical bridge between classical and modern astronomy.

A Practical Next Step

Try these three before moving on to hyperbolas.

1. Find the foci of the ellipse $\frac{x^2}{16} + \frac{y^2}{7} = 1$.

2. Find the area of an ellipse with semi-major axis 6 and semi-minor axis 4.

3. The orbit of Mars has eccentricity ~0.093 and semi-major axis ~228 million km. Estimate the focal distance ($c$).

Also Read:

• Geometry — Concepts, Formulas, Shapes & Topics Guide

• Foci of an Ellipse

• Conic Sections

• Parabola

• Focus of a Parabola

• Directrix of a Parabola

• Equation of a Circle