Directrices of a Hyperbola

Directrices of a Hyperbola

Two parallel lines which are perpendicular to the major axis of a hyperbola. The directrices are between the two parts of a hyperbola and can be used to define it as follows: A hyperbola is the locus of points such that the ratio of the distance to the nearer focus to the distance to the nearer directrix equals a constant that is greater than one. This constant is the eccentricity.

Two diagrams of a hyperbola with foci and directrices labeled. Ratio L1/L2 equals eccentricity, where L1 is distance to focus,...

Key Formula

x = \pm \frac{a}{e} = \pm \frac{a^2}{c}

Where:

$a$ = Semi-transverse axis length — the distance from the center to each vertex along the major axis
$e$ = Eccentricity of the hyperbola, defined as e = c/a, always greater than 1 for a hyperbola
$c$ = Distance from the center to each focus, where c² = a² + b²
$b$ = Semi-conjugate axis length

Worked Example

Problem: Find the equations of the directrices for the hyperbola x²/9 − y²/16 = 1.

Step 1: Identify a² and b² from the standard form. Here a² = 9 and b² = 16, so a = 3 and b = 4.

a = 3, \quad b = 4

Step 2: Calculate c using the relationship c² = a² + b².

c = \sqrt{9 + 16} = \sqrt{25} = 5

Step 3: Find the eccentricity e = c/a.

e = \frac{5}{3}

Step 4: Apply the directrix formula x = ±a/e, or equivalently ±a²/c.

x = \pm \frac{3}{5/3} = \pm \frac{9}{5}

Answer: The directrices are x = 9/5 and x = −9/5 (i.e., x = ±1.8).

Another Example

This example shows a vertical hyperbola, where the directrices are horizontal lines (y = constant) instead of vertical lines. Students must match the directrix orientation to the transverse axis direction.

Problem: Find the directrices of the hyperbola y²/25 − x²/144 = 1.

Step 1: This hyperbola opens vertically because the y² term is positive. Here a² = 25 (under y²) and b² = 144, so a = 5 and b = 12.

a = 5, \quad b = 12

Step 2: Compute c from c² = a² + b².

c = \sqrt{25 + 144} = \sqrt{169} = 13

Step 3: Calculate the eccentricity.

e = \frac{c}{a} = \frac{13}{5}

Step 4: For a vertical hyperbola, the directrices are horizontal lines y = ±a²/c.

y = \pm \frac{25}{13}

Answer: The directrices are y = 25/13 and y = −25/13 (approximately ±1.923).

Frequently Asked Questions

What is the difference between the directrices of a hyperbola and the directrix of a parabola?

A parabola has a single directrix and its eccentricity is exactly 1, meaning the distance to the focus equals the distance to the directrix for every point. A hyperbola has two directrices (one for each branch) and its eccentricity is greater than 1, so the distance to the nearer focus is always greater than the distance to the nearer directrix.

Where are the directrices of a hyperbola located relative to the foci and vertices?

The directrices lie between the center and the vertices, inside the gap between the two branches. The foci, by contrast, lie outside the vertices. In order from the center outward along the transverse axis, the arrangement is: center → directrix → vertex → focus.

How do you use the directrix to verify a point is on a hyperbola?

For any point P on the hyperbola, compute the ratio of its distance to the nearer focus to its distance to the nearer directrix. If this ratio equals the eccentricity e, the point lies on the hyperbola. This focus-directrix property works for all conic sections, not just hyperbolas.

Directrices of a Hyperbola vs. Directrices of an Ellipse

	Directrices of a Hyperbola	Directrices of an Ellipse
Formula	x = ±a²/c (with c² = a² + b²)	x = ±a²/c (with c² = a² − b²)
Eccentricity	e > 1	0 < e < 1
Location relative to vertices	Between the center and the vertices (inside the curve)	Outside the vertices (beyond the curve)
Focus-directrix ratio meaning	Distance to focus > distance to directrix	Distance to focus < distance to directrix
Number of directrices	Two	Two

Why It Matters

Directrices appear in conic sections courses and standardized exams when you need to identify or derive a hyperbola from its focus-directrix property. They provide a unified way to define all conics — parabola, ellipse, and hyperbola — through eccentricity. Understanding directrices also matters in physics and engineering, for example in the reflective properties of hyperbolic mirrors and in satellite navigation systems that use hyperbolic positioning.

Common Mistakes

Mistake: Using c² = a² − b² (the ellipse formula) instead of c² = a² + b² when finding c for a hyperbola.

Correction: For a hyperbola, always use c² = a² + b². The ellipse relationship c² = a² − b² gives a smaller c, which would produce incorrect directrix positions.

Mistake: Writing vertical directrices (x = constant) for a vertical hyperbola, or horizontal directrices for a horizontal hyperbola.

Correction: The directrices are always perpendicular to the transverse axis. A horizontal hyperbola (x² term positive) has vertical directrices x = ±a²/c. A vertical hyperbola (y² term positive) has horizontal directrices y = ±a²/c.

Related Terms

Hyperbola — The conic section these directrices help define
Eccentricity — The constant ratio in the focus-directrix property
Foci of a Hyperbola — The two fixed points paired with the directrices
Major Axis of a Hyperbola — The axis the directrices are perpendicular to
Directrices of an Ellipse — Analogous lines for an ellipse where e < 1
Directrix of a Parabola — Single directrix for the conic with e = 1
Locus — Set of points satisfying the focus-directrix condition
Distance from a Point to a Line — Used to measure distance to the directrix