Hyperbola — Introduction

Think of a sonic boom. A fighter jet flying faster than sound creates a shock wave that expands outward as a cone. If you slice that cone with a horizontal plane, the cross-section you get is not a circle, not an ellipse, not a parabola — it is a hyperbola. Two separate curves, opening away from each other, spreading outward without limit.

You have already met the ellipse, where a point moves so that the sum of its distances from two foci is constant. The hyperbola is the mirror version: a point moves so that the difference of its distances from two foci is constant. The sum locks the point inside a closed oval. The difference lets the point escape to infinity — in two opposite directions. That single sign change, from + to -, transforms the geometry completely.

The hyperbola is the last of the three non-degenerate conic sections you will study (after the parabola and the ellipse). It shows up in the paths of comets that visit the solar system once and never return (they follow hyperbolic orbits around the Sun), in the shape of cooling towers at thermal power plants, and in the mathematics of navigation systems that locate a ship by measuring the time difference of signals from two stations. The two-station navigation problem is literally the definition of a hyperbola.

The definition, precisely

The notation parallels the ellipse:

• a is a positive number. The constant difference of distances is 2a.
• c is the distance from the centre (midpoint of F_1F_2) to each focus. So F_1 and F_2 are at (\pm c, 0) when the hyperbola is centred at the origin with foci on the x-axis.
• The constraint a < c is necessary. If a = c, the locus degenerates to the segment between the foci (on each branch, only the vertex qualifies). If a > c, no point satisfies the condition — the difference of distances from two points can never exceed the distance between the points (by the triangle inequality), so 2a > 2c makes the locus empty.

The absolute value |PF_1 - PF_2| is essential. Without it, the condition PF_1 - PF_2 = 2a (no absolute value) picks out only one of the two branches — the branch closer to F_2. With the absolute value, you get both branches.

Compare with the ellipse: for the ellipse, PF_1 + PF_2 = 2a with a > c. For the hyperbola, |PF_1 - PF_2| = 2a with a < c. Sum versus difference, a > c versus a < c. Everything else follows from this sign change.

Deriving the standard equation

Place the centre at the origin. Put the foci on the x-axis at F_1(-c, 0) and F_2(c, 0). Take any point P(x, y) on the right branch of the hyperbola (so P is closer to F_2). The defining condition, for this branch, is:

Write out the distances:

The derivation follows the same strategy as the ellipse — isolate one radical and square, twice.

Step 1. Isolate one radical.

Why: move the second radical to the right so you can square both sides. Notice the +2a on the right — for the ellipse it was 2a minus the second radical, here it is plus. This is the sign difference that changes everything.

Step 2. Square both sides.

Expand (x + c)^2 = x^2 + 2cx + c^2 and (x - c)^2 = x^2 - 2cx + c^2. The x^2, y^2, and c^2 terms cancel:

Divide by 4:

Why: after expanding and canceling, you are left with one radical isolated on one side. For the ellipse the same step gave a^2 - cx = a\sqrt{(x-c)^2 + y^2}. The sign is flipped because the defining equation had a minus instead of a plus.

Step 3. Square again.

The -2a^2 cx terms cancel:

Why: collect x^2 terms on the left and constants on the right. The expression c^2 - a^2 appears on both sides — and since c > a, this quantity is positive.

Step 4. Define b^2 = c^2 - a^2.

Since c > a, we have b > 0. Substitute:

Divide by a^2 b^2:

Why: dividing by a^2 b^2 normalises the equation to the cleanest possible form. The constant on the right is 1, and the minus sign between the two terms is the hallmark of the hyperbola.

Compare this with the ellipse equation \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1. The only algebraic change is + to -. But geometrically, the curve splits open: the ellipse is a single closed loop, the hyperbola is two separate branches extending to infinity.

The key relation is c^2 = a^2 + b^2 — a plus sign, not the c^2 = a^2 - b^2 of the ellipse. For the ellipse, c < a and the foci are inside the curve. For the hyperbola, c > a and the foci are outside the vertices.

The anatomy of a hyperbola

Every term has a geometric meaning. Here they are, labelled on one picture.

• Centre (C): The midpoint of the segment joining the two foci. In the standard form, it is the origin (0, 0).

• Foci (F_1, F_2): The two fixed points, at (-c, 0) and (c, 0), with c = \sqrt{a^2 + b^2}.

• Transverse axis: The line segment joining the two vertices. Its length is 2a. The vertices are the points where the hyperbola crosses this axis: A(a, 0) and A'(-a, 0).

• Conjugate axis: The line segment of length 2b, perpendicular to the transverse axis through the centre. Its endpoints are B(0, b) and B'(0, -b). These points are not on the hyperbola (substituting (0, b) gives 0 - 1 = -1 \neq 1). They define the rectangle used to draw the asymptotes.

• Semi-transverse axis (a): Half the length of the transverse axis.

• Semi-conjugate axis (b): Half the length of the conjugate axis.

Unlike the ellipse, a and b are not required to satisfy a > b. For a hyperbola, a and b can be in any ratio. The transverse axis is simply the axis along which the curve opens — the one containing the foci and vertices.

The relation c^2 = a^2 + b^2 means: if you draw a right triangle with legs a and b, the hypotenuse is c. For the ellipse, a was the hypotenuse and b, c were the legs. For the hyperbola, c is the hypotenuse. The Pythagorean triple (3, 4, 5) appears in the example above: a = 3, b = 4, c = 5.

Eccentricity

The eccentricity of a hyperbola is the ratio

For a hyperbola, c > a, so e > 1. This is the defining feature that separates it from the ellipse (0 < e < 1) and the parabola (e = 1).

• When e is just slightly greater than 1 (say e = 1.01), the foci are barely outside the vertices, and the hyperbola opens very wide — nearly flat branches, like a very gentle V.
• When e is large (say e = 5), the foci are far from the vertices relative to a, and the branches are narrow — closer to a pair of straight lines emanating from the centre.

Since c^2 = a^2 + b^2, you can write e = \sqrt{1 + b^2/a^2}, or equivalently, b^2 = a^2(e^2 - 1).

Eccentricity and the conic family

The eccentricity e is the single number that classifies every conic section:

As e increases through 1, the closed ellipse opens into a parabola (which opens on one side) and then into a hyperbola (which opens on two sides). The three curves form a continuous family, controlled by a single parameter.

The focus-directrix definition

Like the ellipse and parabola, the hyperbola has a focus-directrix definition.

For the standard hyperbola \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1, each focus has a corresponding directrix:

• Focus F_1(-c, 0), directrix x = -a/e = -a^2/c
• Focus F_2(c, 0), directrix x = a/e = a^2/c

Since e > 1, the directrix x = a/e lies inside the hyperbola (between the two vertices: a/e < a, so the directrix is closer to the centre than the vertex). This is the opposite of the ellipse, where the directrix lies outside.

Verification. Take a point (x, y) on the right branch. Its distance from F_2(c, 0) is \sqrt{(x-c)^2 + y^2}. From the derivation, cx - a^2 = a\sqrt{(x-c)^2 + y^2}, so the distance is (cx - a^2)/a = ex - a. The distance from the directrix x = a/e is x - a/e. The ratio is:

This confirms the equivalence.

The latus rectum

The latus rectum is the chord through a focus perpendicular to the transverse axis. Its computation follows the same pattern as the ellipse.

Take the focus F_2(c, 0). The latus rectum has x = c. Substitute into the hyperbola:

The formula is identical to the ellipse. The derivation is identical too — the difference in signs in the equation does not affect this particular computation, because c^2 - a^2 = b^2 plays the same role as a^2 - c^2 = b^2 in the ellipse derivation.

Worked examples

Common confusions

• "The larger denominator is always a^2." For an ellipse, yes — a > b by convention. For a hyperbola, no. The value a^2 is always the denominator under the positive term, regardless of whether it is larger or smaller than b^2. In \frac{x^2}{9} - \frac{y^2}{16} = 1, we have a = 3 and b = 4, even though b > a.

• "b^2 = a^2 - c^2 for the hyperbola." That is the ellipse relation. For the hyperbola, b^2 = c^2 - a^2. The mnemonic: the ellipse has c < a (foci inside the curve), so b^2 = a^2 - c^2 is positive. The hyperbola has c > a (foci outside the vertices), so you need b^2 = c^2 - a^2 to stay positive.

• "The points (0, \pm b) are on the hyperbola." They are not. Substituting (0, b) into \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 gives 0 - 1 = -1 \neq 1. The points (0, \pm b) define the conjugate axis but do not lie on the curve. This is different from the ellipse, where the co-vertices (0, \pm b) are on the curve.

• "Eccentricity close to 1 means the hyperbola is nearly a straight line." Eccentricity close to 1 (from above) means b is close to 0, so the hyperbola is very wide and flat — the branches barely curve away from the transverse axis. A large eccentricity (say e = 10) gives narrow branches. This is the opposite of the intuition many students have.

Going deeper

The rectangular hyperbola

When a = b, the hyperbola \frac{x^2}{a^2} - \frac{y^2}{a^2} = 1 simplifies to x^2 - y^2 = a^2. Its eccentricity is e = \sqrt{1 + 1} = \sqrt{2}. This is called a rectangular hyperbola because its asymptotes (the lines y = \pm x) are perpendicular to each other.

If you rotate a rectangular hyperbola by 45°, its equation becomes xy = k for some constant k. The familiar curve y = 1/x — the hyperbola from the derivative article — is a rectangular hyperbola with k = 1. So the function you have been differentiating since you first met derivatives is a conic section in disguise.

The conic family, unified

All three non-degenerate conics — ellipse, parabola, hyperbola — can be described by a single equation using the focus-directrix definition:

• e < 1: ellipse (closed curve, two foci, sum of distances constant)
• e = 1: parabola (open curve, one focus, equidistant from focus and directrix)
• e > 1: hyperbola (two branches, two foci, difference of distances constant)

The parabola is the boundary case. As e approaches 1 from below, the ellipse stretches longer and longer until one end escapes to infinity — becoming a parabola. As e increases past 1, the curve "breaks" into two pieces — the parabola's single branch splits into the hyperbola's two branches. The entire family is controlled by this single number e.

Bhaskara II, in the 12th-century Siddhanta Shiromani, worked with the geometry of planetary orbits that we now recognise as conic sections. The unifying role of eccentricity, linking circles to ellipses to parabolas to hyperbolas, was not formalised until later — but the geometric intuition that these curves form a family was present in Indian astronomical traditions long before the modern algebraic framework.

The two-focus construction

For the ellipse, you used the "string and pins" construction: two pins (foci), a loop of string, and a pencil tracing all points where the sum of distances is constant. For the hyperbola, the analogous construction uses a difference of distances — harder to do with string, but possible with a mechanical device.

Take two pins at F_1 and F_2. Attach a ruler of length \ell to F_1 at one end, so it can rotate freely. Tie a string of length \ell - 2a from the other end of the ruler to F_2. Now hold a pencil where the string meets the ruler, keeping the string taut. As the ruler rotates, the pencil traces a curve. At each position, the pencil is at distance r_1 from F_1 (along the ruler) and r_2 from F_2 (along the string). The ruler has length \ell = r_1 + (\ell - 2a - r_2 + r_2)... more simply: the string length is \ell - 2a, so r_2 = \ell - r_1 - (\ell - 2a - r_2). The constraint is r_1 - r_2 = 2a, which traces one branch of the hyperbola.

This is a physical construction, not just algebra. If you have a ruler, string, and two pins, you can draw a hyperbola on a board — the same way you drew an ellipse, but with the defining condition changed from sum to difference.

Where this leads next

You now know what a hyperbola is, where its equation comes from, and what each term means. The next articles develop the tools for working with it:

• Hyperbola — Conjugate and Rectangular — the conjugate hyperbola, the rectangular hyperbola xy = c^2, and their properties.
• Hyperbola — Asymptotes — the two lines that the branches approach but never reach, and why they make the hyperbola unique among conics.
• Ellipse — Introduction — revisit the ellipse if you want to compare the two curves side by side; the sign changes are all spelled out there.
• Coordinate Geometry Basics — if the distance formula or locus definition felt unfamiliar.
• Locus — the general idea of a curve defined by a geometric condition, which is exactly how the hyperbola is defined.