The ellipse

What defines an ellipse

• An ellipse is the set of points whose distances to two fixed points (the foci) add to a constant.
• The gardener's construction makes this concrete: pin the two ends of a loose string, pull it taut with a pencil, and the pencil traces an ellipse.
• It is a closed, bounded curve — unlike the parabola and hyperbola, which run off to infinity.
• A circle is the special case where the two foci coincide at the centre.

The standard equation

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

• Each part:
  • $(h, k)$ — the centre
  • $a$ — the distance from the centre to the curve along the $x$-direction (the semi-axis in $x$)
  • $b$ — the distance from the centre to the curve along the $y$-direction
• The right-hand side must be 1. If it is not, divide the whole equation through before reading anything off.
• The plus sign identifies the ellipse. With a minus it would be a hyperbola; with $a = b$ it is a circle.

Major and minor axes

• The major axis is the longer one, along the direction of the larger denominator. The minor axis is the shorter one.
• The endpoints of the major axis are the vertices; the endpoints of the minor axis are the co-vertices.
• Full lengths are $2a$ and $2b$ — the values $a$ and $b$ themselves are only half-axes.
• The larger denominator tells you the orientation:
  • $a > b$ → the ellipse is wider than it is tall; major axis horizontal
  • $b > a$ → taller than wide; major axis vertical

The foci

• The foci lie on the major axis, at distance $c$ from the centre, where

$$c^2 = (\text{larger semi-axis})^2 - (\text{smaller semi-axis})^2$$

• For a horizontal major axis ($a > b$): $c^2 = a^2 - b^2$, foci at $(h \pm c,\ k)$.
• For a vertical major axis ($b > a$): $c^2 = b^2 - a^2$, foci at $(h,\ k \pm c)$.
• Subtract, and always the smaller from the larger. A negative $c^2$ means you have subtracted the wrong way round.
• The constant sum of distances is $2 \times$ the semi-major axis. For a horizontal ellipse, every point on the curve satisfies $PF_1 + PF_2 = 2a$.

Eccentricity

• Eccentricity measures how "squashed" an ellipse is:

$$e = \frac{c}{\text{semi-major axis}}, \qquad 0 \leq e < 1$$

• $e = 0$ — a circle, with both foci at the centre.
• $e$ close to 1 — a long, thin ellipse with the foci near the ends.
• Earth's orbit has $e \approx 0.017$, which is why it is very nearly circular; Halley's Comet has $e \approx 0.967$.

Completing the square for an ellipse

• The general form has both variables squared with the same sign but different coefficients:

$$9x^2 + 25y^2 - 18x + 100y - 116 = 0$$

• The steps:
  • Group the $x$ terms and the $y$ terms.
  • Factor out the coefficient of each squared term before completing the square.
  • Complete the square inside each bracket, and add the correct amount — multiplied by the factor outside — to the right-hand side.
  • Divide through to make the right-hand side 1.
• Worked through:
  • $$9(x^2 - 2x) + 25(y^2 + 4y) = 116$$
  • $$9(x^2 - 2x + 1) + 25(y^2 + 4y + 4) = 116 + 9(1) + 25(4)$$
  • $$9(x-1)^2 + 25(y+2)^2 = 116 + 9 + 100 = 225$$
  • Divide by 225: $\dfrac{(x-1)^2}{25} + \dfrac{(y+2)^2}{9} = 1$
  • Centre $(1, -2)$, $a = 5$, $b = 3$, major axis horizontal
• The factor outside multiplies what you add to the right. Adding 1 inside a bracket multiplied by 9 adds 9 to the equation, not 1. This is the single biggest source of error in the topic.

The reflective property

• A ray from one focus reflects off the ellipse and passes through the other focus.
• This is why a whispering gallery works: a whisper at one focus is heard clearly at the other, however large the room.
• It is also the principle behind lithotripsy, where shock waves generated at one focus are concentrated on a kidney stone placed at the other.