The Ellipse

Take a loop of string, pin its two ends to a board, hook a pencil inside the loop and pull it taut. Now drag the pencil all the way round, keeping the string tight the whole time. The curve it traces is not a circle — it is a squashed, egg-smooth oval called an ellipse. This is the gardener's trick for marking out an elliptical flower bed, and it hides the exact definition of the shape.

Because the string never changes length, the pencil obeys one rule at every instant: its distance to the first pin plus its distance to the second pin is always the same. Those two pins are the foci (singular focus), and the constant total is the ellipse's defining promise.

An ellipse is every point P for which the sum of the distances to two fixed points — the foci F_1 and F_2 — is a constant.
That constant is written 2a: PF_1 + PF_2 = 2a.
Compare it with a circle, where a single distance stays constant — an ellipse simply uses two anchor points instead of one.

The anatomy of an ellipse

Centre the ellipse on the origin and lay its longest direction along the x-axis. Three lengths name every part of it:

the semi-major axis a — half the long way across, reaching the vertices at (\pm a, 0);
the semi-minor axis b — half the short way, reaching the co-vertices at (0, \pm b);
the focal distance c — from the centre out to each focus at (\pm c, 0).

Step through the figure below: first the oval and its foci, then the two half-axes a and b, then a roving point P whose two focal radii always sum to the same 2a.

The equation, and how the three lengths connect

Feed the "sum of distances is 2a" rule through the distance formula and tidy up, and the whole curve collapses to one clean equation:

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

Set y = 0 and you get x = \pm a (the vertices); set x = 0 and you get y = \pm b (the co-vertices). And the foci? Send P to the top co-vertex (0, b): by symmetry each focal radius is equal, so each is a, and the right triangle with legs b and c and hypotenuse a gives Pythagoras:

The foci sit at (\pm c, 0) with c^2 = a^2 - b^2.
So c = \sqrt{a^2 - b^2} — always smaller than a, so the foci live inside the ellipse.
Note the minus sign: this is not the circle's a^2 + b^2. (For a hyperbola the sign flips again.)

Eccentricity: how squashed is it?

One number captures an ellipse's shape independent of its size — the eccentricity:

e = \frac{c}{a}, \qquad 0 \le e < 1.

Because c < a, eccentricity always lands between 0 and 1. When the foci merge at the centre, c = 0 and e = 0 — a perfect circle. As the foci slide apart toward the vertices, e \to 1 and the ellipse stretches into a long, thin cigar. So eccentricity is a "how oval" dial: 0 is round, close to 1 is very elongated.

Worked examples

1 · Read everything off the equation. Describe \dfrac{x^2}{25} + \dfrac{y^2}{9} = 1.

The bigger denominator is 25 and it is under x^2, so the major axis is horizontal and a^2 = 25 \Rightarrow a = 5. The other gives b^2 = 9 \Rightarrow b = 3. Then

c = \sqrt{a^2 - b^2} = \sqrt{25 - 9} = \sqrt{16} = 4,

so the vertices are (\pm 5, 0), the co-vertices (0, \pm 3), the foci (\pm 4, 0), and the eccentricity e = \dfrac{c}{a} = \dfrac{4}{5} = 0.8 — a noticeably squashed ellipse.

2 · Build the equation from its features. Find the ellipse with vertices (\pm 6, 0) and foci (\pm 4, 0).

The vertices give a = 6, so a^2 = 36. The foci give c = 4. Rearrange the focal relation for b^2:

b^2 = a^2 - c^2 = 36 - 16 = 20.

Everything is horizontal (both features lie on the x-axis), so the equation is

\frac{x^2}{36} + \frac{y^2}{20} = 1,

with eccentricity e = \dfrac{4}{6} = \dfrac{2}{3} \approx 0.67.

In 1609 Kepler shattered two thousand years of "orbits must be perfect circles": his first law says every planet travels on an ellipse with the Sun at one focus (the other focus is just empty space). Earth's orbit is only gently oval — eccentricity about 0.017, almost a circle — while a comet can swing on an ellipse so stretched that e is a whisker under 1.

The two foci also give the ellipse a magic acoustic trick. Any ray leaving one focus — light or sound — bounces off the curve and passes exactly through the other focus. That is why a whispering gallery works: stand at one focus of an elliptical dome, murmur, and a friend at the far focus hears you clearly while everyone between hears nothing. The same reflection property focuses shock waves in a lithotripter to shatter kidney stones without surgery.

The letter a is always the semi-major axis — the root of the larger denominator — and b the smaller. Do not assume a^2 always sits under x^2. Read which number is bigger:

If the larger denominator is under x^2, the major axis is horizontal and the foci are on the x-axis at (\pm c, 0).
If the larger denominator is under y^2 — as in \dfrac{x^2}{9} + \dfrac{y^2}{25} = 1 — the major axis is vertical, so a^2 = 25, b^2 = 9, and the foci are on the y-axis at (0, \pm c).

The relation c^2 = a^2 - b^2 never changes — just always feed it the larger value as a^2. If you ever get a negative under the square root, you mixed up a and b.