Hyperbola

A hyperbola is one of the four classical conic sections, formed when a plane cuts through both nappes (cones) of a double cone at an angle steeper than the side of the cone. Unlike the ellipse, which is a closed curve, a hyperbola consists of two separate open branches that curve away from each other symmetrically. Hyperbolas appear throughout mathematics, physics, and engineering — from the paths of comets under repulsive gravitational forces, to the shape of cooling towers, to the design of hyperbolic mirrors in telescopes. The defining geometric property of a hyperbola is that for any point P on the curve, the absolute difference of the distances from P to two fixed points called the foci is constant. If the foci are F₁ and F₂, then |PF₁ − PF₂| = 2a, where a is the semi-transverse axis. This 'constant difference' definition contrasts with the ellipse, where the sum of distances is constant. This seemingly small change — sum versus difference — produces a dramatically different shape with two branches opening in opposite directions. Mastering the hyperbola means understanding its standard equation, its key measurements (semi-transverse axis a, semi-conjugate axis b, and focal distance c), and derived quantities like eccentricity and the latus rectum. These parameters completely describe the shape and size of any hyperbola and allow you to sketch it, find its asymptotes, and solve real problems in coordinate geometry.

Key takeaways
  • The Standard Equation (Horizontal Transverse Axis)
  • The Standard Equation (Vertical Transverse Axis)
  • Asymptotes
  • Eccentricity
  • Latus Rectum
  • Directrices

The Standard Equation (Horizontal Transverse Axis)

When the transverse axis lies along the x-axis and the centre is at the origin, the standard form is x²/a² − y²/b² = 1. The vertices are at (±a, 0) and the foci are at (±c, 0), where c² = a² + b². Note that for a hyperbola, c > a, so c² = a² + b² (compare with the ellipse where c² = a² − b²). The two branches open left and right.

The Standard Equation (Vertical Transverse Axis)

When the transverse axis lies along the y-axis, the equation becomes y²/a² − x²/b² = 1. Vertices are now at (0, ±a) and foci at (0, ±c). The branches open upward and downward. Always identify which variable has the positive term — that variable's axis is the transverse axis.

Asymptotes

Every hyperbola has two asymptotes — straight lines that the branches approach but never touch. For x²/a² − y²/b² = 1, the asymptotes are y = ±(b/a)x. For y²/a² − x²/b² = 1, they are y = ±(a/b)x. The asymptotes pass through the centre and form a visual 'guide' for sketching the branches. The combined equation of the asymptotes can be written as x²/a² − y²/b² = 0.

Eccentricity

Eccentricity e measures how 'open' or 'spread' the hyperbola is. It is defined as e = c/a. Since c > a for any hyperbola, we always have e > 1. A hyperbola with e only slightly above 1 has narrow branches; as e increases, the branches open wider. Compare: for an ellipse 0 < e < 1, for a parabola e = 1, and for a hyperbola e > 1. Eccentricity is dimensionless and independent of the actual size of the curve.

Latus Rectum

The latus rectum is the chord of the hyperbola that passes through a focus and is perpendicular to the transverse axis. To find its length, substitute x = c into the equation x²/a² − y²/b² = 1: y² = b²(c²/a² − 1) = b²·b²/a² = b⁴/a². So y = ±b²/a, and the full length of the latus rectum is 2b²/a. Each focus has its own latus rectum of the same length. The latus rectum is important because it determines how quickly the branches spread near the focus.

Directrices

Each branch of a hyperbola has an associated directrix — a vertical (or horizontal) line such that the ratio of any point's distance to the nearest focus and its distance to the corresponding directrix equals e. The directrices for the horizontal hyperbola are x = ±a/e = ±a²/c. This focus-directrix property provides an alternative definition of the hyperbola using eccentricity.

The Key Equation

x2a2−y2b2=1,c2=a2+b2,e=ca>1,ℓ=2b2a\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1, \quad c^2 = a^2 + b^2, \quad e = \frac{c}{a} > 1, \quad \ell = \frac{2b^2}{a}
  • a — Semi-transverse axis (length units): Half the length of the transverse axis; the distance from the centre to each vertex. The two vertices are the closest points on the hyperbola to the centre.
  • b — Semi-conjugate axis (length units): Half the length of the conjugate axis. It determines the slopes of the asymptotes (±b/a) but is NOT the distance from the centre to any point on the hyperbola.
  • c — Focal distance (length units): Distance from the centre to each focus. For a hyperbola, c² = a² + b², so c is always greater than a.
  • e — Eccentricity (dimensionless): The ratio c/a, always greater than 1 for a hyperbola. It measures how open the branches are: larger e means wider-opening branches.
  • \ell — Latus rectum (full length) (length units): The length of the chord through a focus perpendicular to the transverse axis. Equal to 2b²/a, analogous to the same formula for an ellipse.

Worked Example

Problem. For the hyperbola 9x² − 16y² = 144, find: (i) the values of a, b, and c; (ii) the eccentricity; (iii) the length of the latus rectum; (iv) the equations of the asymptotes.

a=4,  b=3,  c=5,  e=54,  ℓ=92,  y=±34xa=4,\; b=3,\; c=5,\; e=\tfrac{5}{4},\; \ell=\tfrac{9}{2},\; y=\pm\tfrac{3}{4}x

Answer. a = 4, b = 3, c = 5; eccentricity e = 5/4; latus rectum length = 9/2; asymptotes y = ±(3/4)x

Figure
Diagram
xyy=¾xy=−¾x(a,0)(−a,0)F₂(c,0)F₁(−c,0)Latusrectum= 2b²/aHyperbola: x²/a² − y²/b² = 1Transverse axis (2a)
Origins · 1 minApollonius of Perga and how this was discovered
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Apollonius of Perga
-262–-190 · Greek (Ancient)

Apollonius of Perga, known as 'The Great Geometer', wrote the monumental eight-volume treatise Conics (Κωνικά) around 200 BCE. He was the first to give the conic sections their modern names — ellipse, parabola, and hyperbola — and systematically derived their properties purely from geometry. His name 'hyperbola' comes from the Greek ὑπερβολή (hyperbolē), meaning 'excess', referring to the fact that the angle of the cutting plane exceeds that of the cone's side. Apollonius studied the double-branched nature of the hyperbola and its asymptotes, work that remained the definitive reference on conics for nearly 1,800 years until the Renaissance. Apollonius's Conics originally had eight books, but only seven survive — the eighth was lost in antiquity. Mathematicians have tried for centuries to reconstruct its contents from references in the surviving books.

  • Ellipse: Equation, Eccentricity and Latus Rectum — The ellipse is the 'sister' conic to the hyperbola — both share the definition using two foci, but the ellipse uses the sum of distances (giving e < 1) while the hyperbola uses the difference (giving e > 1). The formulas for latus rectum 2b²/a and eccentricity c/a are structurally identical, making direct comparison very instructive.
  • Parabola: Focus, Directrix and Standard Equation — The parabola is the boundary case between the ellipse and hyperbola, with eccentricity exactly equal to 1. Understanding how eccentricity varies across all three conics (0 < e < 1 for ellipse, e = 1 for parabola, e > 1 for hyperbola) gives a unified view of all conic sections.
  • General Second-Degree Equation and Identification of Conics — Any conic section, including a hyperbola, can be written as Ax² + Bxy + Cy² + Dx + Ey + F = 0. Learning the discriminant condition B² − 4AC > 0 to identify a hyperbola (versus < 0 for ellipse and = 0 for parabola) connects the standard forms to the general equation.
  • Rectangular Hyperbola: xy = k — The rectangular hyperbola xy = k is a special case where the asymptotes are perpendicular (along the coordinate axes). It has eccentricity e = √2 and appears in inverse proportion graphs. Recognising it as a rotated version of x² − y² = constant deepens understanding of hyperbola geometry.