Foci of a Hyperbola — Definition, Formula & Examples

Foci of a Hyperbola

Two fixed points located inside each curve of a hyperbola that are used in the curve's formal definition. A hyperbola is defined as follows: For two given points, the foci, a hyperbola is the locus of points such that the difference between the distance to each focus is constant.

Hyperbola diagram with equation (x-h)²/a²-(y-k)²/b²=1, showing center, foci, vertices, and asymptote box where a²+b²=c².

Hyperbola with two foci, points L1 and L2 marked, point P on curve. |L1−L2|=2a (horizontal), |L1−L2|=2b (vertical).

Key Formula

c^2 = a^2 + b^2

Where:

$c$ = Distance from the center of the hyperbola to each focus
$a$ = Distance from the center to each vertex (semi-transverse axis length)
$b$ = Semi-conjugate axis length

Worked Example

Problem: Find the foci of the hyperbola given by the equation x²/9 − y²/16 = 1.

Step 1: Identify a² and b² from the standard form. The equation is already in the form x²/a² − y²/b² = 1.

a^2 = 9, \quad b^2 = 16

Step 2: Use the relationship c² = a² + b² to find c.

c^2 = 9 + 16 = 25

Step 3: Take the square root to get c.

c = \sqrt{25} = 5

Step 4: Since the x²-term is positive, the transverse axis is horizontal. The foci lie on the x-axis at (±c, 0).

\text{Foci: } (5,\, 0) \text{ and } (-5,\, 0)

Answer: The foci are at (5, 0) and (−5, 0).

Another Example

This example differs by having a vertical transverse axis and a center that is not at the origin, requiring you to apply shifts to the focus coordinates.

Problem: Find the foci of the hyperbola given by (y − 2)²/25 − (x + 3)²/144 = 1.

Step 1: Identify the center from the shifted form. The center is at (h, k) = (−3, 2).

h = -3, \quad k = 2

Step 2: Identify a² and b². Because the y-term comes first (is positive), the transverse axis is vertical, with a² = 25 and b² = 144.

a^2 = 25, \quad b^2 = 144

Step 3: Calculate c using c² = a² + b².

c^2 = 25 + 144 = 169 \implies c = 13

Step 4: Since the transverse axis is vertical, the foci are displaced vertically from the center by ±c.

\text{Foci: } (-3,\, 2 + 13) = (-3,\, 15) \quad \text{and} \quad (-3,\, 2 - 13) = (-3,\, -11)

Answer: The foci are at (−3, 15) and (−3, −11).

Frequently Asked Questions

How do you find the foci of a hyperbola?

Start by writing the equation in standard form to identify a² and b². Then compute c using c² = a² + b². If the transverse axis is horizontal (x²-term is positive), the foci are at (h ± c, k). If the transverse axis is vertical (y²-term is positive), the foci are at (h, k ± c), where (h, k) is the center.

What is the difference between the foci of a hyperbola and the foci of an ellipse?

For an ellipse, c² = a² − b², so c is always less than a and the foci sit between the vertices. For a hyperbola, c² = a² + b², so c is always greater than a and the foci lie beyond the vertices. Additionally, an ellipse uses the sum of distances to the foci (constant), while a hyperbola uses the absolute difference of distances (constant).

Why does a hyperbola have two foci?

The two foci are essential to the geometric definition of a hyperbola: a point lies on the hyperbola if and only if the absolute difference of its distances to the two foci equals 2a. With only one focus, you could not define this distance-difference relationship, and you would get a different conic (like a parabola, which has one focus).

Foci of a Hyperbola vs. Foci of an Ellipse

	Foci of a Hyperbola	Foci of an Ellipse
Key formula	c² = a² + b²	c² = a² − b²
Distance property	\|d₁ − d₂\| = 2a (difference is constant)	d₁ + d₂ = 2a (sum is constant)
Foci position relative to vertices	Foci lie outside the vertices (c > a)	Foci lie inside the vertices (c < a)
Number of branches	Two separate branches, one focus per branch	One closed curve containing both foci

Why It Matters

You encounter the foci of a hyperbola in precalculus and analytic geometry courses whenever you classify or graph conic sections. They are also central in physics and engineering — for example, certain reflective properties of hyperbolas depend on the foci, and hyperbolic navigation systems (like LORAN) use the distance-difference definition directly. Understanding foci also builds the foundation for studying eccentricity and the directrix form of conics.

Common Mistakes

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

Correction: Remember that for a hyperbola, c is the largest of the three values a, b, and c. The formula is c² = a² + b², which always gives c > a.

Mistake: Placing the foci along the wrong axis — for example, putting them on the x-axis when the transverse axis is actually vertical.

Correction: The foci always lie along the transverse axis. Check which variable's term is positive in the standard form: if the x²-term is positive, the transverse axis is horizontal; if the y²-term is positive, it is vertical.

Related Terms

Hyperbola — The conic curve defined by its foci
Focus (conic section) — General concept of a focus for any conic
Foci of an Ellipse — Analogous concept using sum of distances
Focus of a Parabola — Single focus defining a parabola
Directrices of a Hyperbola — Lines paired with foci to define eccentricity
Focal Radius — Distance from a focus to a point on the curve
Locus — Set of points satisfying the distance condition
Constant — The fixed difference 2a in the definition