The Ellipse

Take a piece of string, pin its two ends to a table, and trace a curve while keeping the string taut. What you draw is an ellipse. The two pins are the foci; the curve is the locus of points where the sum of distances to the foci stays constant (equal to the string length). This pinned-string construction is more than a party trick, it is the geometric definition that explains Kepler's discovery that planets orbit the Sun in ellipses, the Sun sitting at one focus.

The standard equation

An ellipse centred at the origin with semi-axes $a$ along $x$ and $b$ along $y$ (with $a, b > 0$) satisfies $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$. If $a > b$, the ellipse is wider than tall and the major axis runs along the $x$-axis with length $2a$; the minor axis runs along $y$ with length $2b$. The vertices and co-vertices are $(\pm a, 0)$ and $(0, \pm b)$. If $b > a$, swap roles.

Foci and eccentricity

The foci sit on the major axis at distance $c$ from the centre, where $c = \sqrt{a^2 - b^2}$ (foci at $(\pm c, 0)$). The string length, the sum of distances from any point on the ellipse to the two foci, equals $2a$. The eccentricity is $e = c/a$ with $0 \leq e < 1$. When $e = 0$ the foci coincide at the centre and the ellipse is a circle. As $e \to 1$ the ellipse stretches. Earth's orbit has $e \approx 0.017$; Halley's comet has $e \approx 0.97$.

• Astronomy: Planetary orbits are ellipses with the Sun at one focus (Kepler's first law); the eccentricity tells you how "oval" the orbit is, Earth's is 0.017, Mercury's 0.206, Halley's comet 0.967.
• Architectural Acoustics: An elliptical chamber, like Statuary Hall in the US Capitol or the Mormon Tabernacle in Salt Lake City, reflects sound from one focus precisely to the other; standing at one focus, you hear a whisper from the other across the room.
• Medicine: Lithotripsy uses an ellipsoidal water bath to focus shock waves: kidney stones at one focus are pulverised by sound waves originating at the other focus.

(Select "ellipse" mode. Slide $a$ and $b$; watch the foci move as eccentricity changes.)

Shifted ellipses

An ellipse centred at $(h, k)$ satisfies $(x - h)^2/a^2 + (y - k)^2/b^2 = 1$. The foci shift to $(h \pm c, k)$. To find the centre of an ellipse given in expanded form, complete the square in both $x$ and $y$.

Try it

• Find the foci and eccentricity of $x^2/25 + y^2/9 = 1$.
• An ellipse has foci at $(\pm 3, 0)$ and vertices at $(\pm 5, 0)$. Find its equation.
• For an ellipse with $a = 6, c = 4$, what is $b$?

Pause: if $a = b$, then $c = 0$, foci merge at the centre, and the ellipse is a circle.

A trap to watch for

The sign in $c^2 = a^2 - b^2$ uses subtraction, but for the hyperbola the analogous formula uses addition: $c^2 = a^2 + b^2$. Memory device: an ellipse is contained, foci inside the bounding rectangle, $c < a$. A hyperbola is unbounded, foci outside, $c > a$. A second trap: check which semi-axis is larger. If the equation is $x^2/16 + y^2/49 = 1$, the major axis is along $y$.

What you now know

You can write the standard ellipse equation, identify major/minor axes, locate the foci via $c = \sqrt{a^2 - b^2}$, compute eccentricity, and recognise a circle as the $e = 0$ special case. The next section turns the sum-of-distances into a difference-of-distances and produces the hyperbola.

Quick check

Mark section complete →

References

• Lang, S. (1971). Basic Mathematics. Springer. Chapter 12, §3.
• Apostol, T. M. (1969). Calculus, Volume 2. Wiley. Chapter 13: conic sections and Kepler's laws.
• Hartshorne, R. (2000). Geometry: Euclid and Beyond. Springer. Chapter 6: Apollonius's treatment.