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Ellipses and Hyperbolas: Two Foci

The parabola needed one focus; the ellipse and hyperbola each need two. One curve fixes the sum of the distances to its foci, the other fixes the difference — and from those two stubborn rules every property, from the gardener's string to the comet's path, quietly follows.

From one focus to two

In the last guide the parabola was built from a single special point — its focus — balanced against a line, the directrix. That balance gave a curve that opens forever in one direction. Now we make a small but powerful change: instead of one focus and a line, we take two foci, two fixed points F_1 and F_2, and we govern a moving point P entirely by its two distances |P F_1| and |P F_2|. Everything in this guide flows from that single decision — two anchor points, and one rule tying the two distances together.

There are two honest ways to tie two distances together with a constant, and each gives a different curve. If we demand that the sum |P F_1| + |P F_2| stay constant, P traces an ellipse — a closed, oval loop. If instead we demand that the difference | |P F_1| - |P F_2| | stay constant, P traces a hyperbola — a pair of open branches flying apart. Sum versus difference: that is the whole distinction, and it is worth holding onto, because almost everyone confuses which is which at first.

The ellipse: a constant sum, and the gardener's string

The focal-distance definition of the ellipse says: an ellipse is the set of points P for which |P F_1| + |P F_2| equals a fixed constant, call it 2*a, that is larger than the distance |F_1 F_2| between the foci. This rule has a beautifully physical model. Press two pins into a board at F_1 and F_2, loop a slack string of length 2*a around them, pull it taut with a pencil tip, and slide the pencil all the way around. The taut string forces |P F_1| + |P F_2| to stay equal to the string's length at every instant, so the pencil draws a perfect ellipse. Gardeners use exactly this trick — two stakes and a rope — to lay out oval flower beds.

Place the foci symmetrically on the x-axis at (-c, 0) and (c, 0) and the algebra becomes clean. The points farthest left and right, the vertices, sit at (-a, 0) and (a, 0); the segment between them is the major axis, of length 2*a. The points at the top and bottom, (0, -b) and (0, b), give the minor axis of length 2*b. A short computation with the constant-sum rule at a top vertex (where the two focal distances are equal) yields the single most important relation for the ellipse: a^2 = b^2 + c^2. From it the standard equation falls out, x^2/a^2 + y^2/b^2 = 1.

ellipse, foci at (-c,0) and (c,0):

   |P F_1| + |P F_2| = 2a        (constant sum, 2a > 2c)
   a^2 = b^2 + c^2               (a = semi-major, b = semi-minor, c = focal half-distance)
   x^2/a^2 + y^2/b^2 = 1         (standard equation, a > b)

   a circle is the special case c = 0, F_1 = F_2, a = b
The ellipse's three facts: the constant-sum rule, the a-b-c relation, and the equation it produces.

The hyperbola: a constant difference, and two branches

Swap the sum for a difference and the closed loop bursts open. The hyperbola is the set of points P for which the absolute difference | |P F_1| - |P F_2| | equals a fixed constant 2*a, this time smaller than |F_1 F_2|. The absolute value is doing real work: points closer to F_1 satisfy |P F_1| - |P F_2| = -2*a and form one branch; points closer to F_2 satisfy |P F_1| - |P F_2| = +2*a and form the other. So a hyperbola always comes in two pieces, one hugging each focus, mirror images across the midpoint. The vertices, again the points where the curve turns, sit at (-a, 0) and (a, 0) when the foci lie on the x-axis at (-c, 0) and (c, 0).

The hyperbola carries one feature the ellipse lacks: two straight lines it forever approaches but never touches, its asymptotes. Far from the centre the two branches straighten out and run almost exactly along the lines y = (b/a)*x and y = -(b/a)*x, where now b is defined by c^2 = a^2 + b^2 — notice the sign flips from the ellipse's a^2 = b^2 + c^2, because here c is the largest of the three. The standard equation also flips a sign: x^2/a^2 - y^2/b^2 = 1. That single minus sign, against the ellipse's plus, is the algebraic signature of 'difference instead of sum.'

Why the foci matter: reflection and the sky

The foci are not bookkeeping points; they have a striking optical life, the reflective property of conics. In an ellipse, a ray of light or sound leaving one focus reflects off the curve and passes exactly through the other focus — every path, no matter which way it started. This is why a 'whispering gallery' works: stand at one focus of an elliptical room, murmur, and a friend at the other focus hears you clearly while everyone between hears nothing. The hyperbola has the mirror-image version: a ray aimed at one focus bounces off the near branch as though it had come straight from the other focus.

This is also why the conics rule the heavens. Under gravity, a small body orbiting a large one moves on a conic with the large mass sitting at a focus — not at the centre. A bound planet or comet rides an ellipse, the Sun at one focus; an unbound comet that visits once and leaves forever rides a hyperbola, again with the Sun at a focus. The two-focus geometry you built from a string and a rule is the same geometry Kepler read off the night sky, and it is the eccentricity — the next guide's single number — that decides which conic an orbit is.

A worked example, end to end

Let us pin down a concrete ellipse and read off everything. Suppose the foci are at (-3, 0) and (3, 0), so c = 3, and the constant sum is 2*a = 10, so a = 5. The steps below show how the focal definition alone hands you the full equation, the shape, and the vertices — no memorized formula required beyond the a-b-c relation, which itself came from the definition.

  1. Read off the constants from the focal data: c = 3 (half the distance between foci) and a = 5 (half the constant sum 2*a = 10). Check it is a real ellipse: 2*a = 10 is bigger than |F_1 F_2| = 6, so the string has slack. Good.
  2. Find b from the ellipse relation a^2 = b^2 + c^2: b^2 = 25 - 9 = 16, so b = 4.
  3. Write the standard equation: x^2/25 + y^2/16 = 1.
  4. Locate the features: vertices at (-5, 0) and (5, 0) on the major axis, minor-axis ends at (0, -4) and (0, 4).
  5. Sanity-check the definition at a vertex: from (5, 0), the distances to the foci are |5 - (-3)| = 8 and |5 - 3| = 2, and 8 + 2 = 10 = 2*a. The constant sum holds, exactly as designed.

Run the same machine for a hyperbola and the only changes are signs: with foci at (-5, 0), (5, 0) and constant difference 2*a = 6, you would get c = 5, a = 3, then b^2 = c^2 - a^2 = 25 - 9 = 16, giving x^2/9 - y^2/16 = 1 and asymptotes y = (4/3)*x and y = -(4/3)*x. Same three steps, sum traded for difference, plus traded for minus — that symmetry between the two curves is the real prize of meeting them together.