Hyperbola

Hyperbola

A conic section that can be thought of as an inside-out ellipse.

Formally, a hyperbola can be defined as follows: For two given points, the foci, a hyperbola is the locus of points such that the difference between the distances to each focus is constant.

Two hyperbola graphs: horizontal ((x-h)²/a² - (y-k)²/b² = 1) and vertical ((y-k)²/b² - (x-h)²/a² = 1), with center (h,k), axes...

Hyperbola diagram showing two curves on x-y axes with box, foci, and labels: (x-h)²/a² - (y-k)²/b² = 1; a²+b²=c²

Hyperbola with two foci showing L1 and L2 distances from point P; |L1−L2|=2a (horizontal) or 2b (vertical).

Key Formula

\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1

Where:

$a$ = The distance from the center to each vertex (half the transverse axis length)
$b$ = The distance related to the conjugate axis; determined by b² = c² − a²
$c$ = The distance from the center to each focus, where c² = a² + b²
$x, y$ = Coordinates of any point on the hyperbola (centered at the origin)

Worked Example

Problem: Find the foci, vertices, and asymptotes of the hyperbola given by x²/9 − y²/16 = 1.

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

a^2 = 9,\quad b^2 = 16 \implies a = 3,\; b = 4

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

c^2 = 9 + 16 = 25 \implies c = 5

Step 3: The vertices lie on the transverse axis (x-axis) at distance a from the center.

\text{Vertices: } (\pm 3,\, 0)

Step 4: The foci lie on the transverse axis at distance c from the center.

\text{Foci: } (\pm 5,\, 0)

Step 5: The asymptotes of a hyperbola in this form are the lines y = ±(b/a)x.

y = \pm\frac{4}{3}x

Answer: Vertices at (±3, 0), foci at (±5, 0), and asymptotes y = ±(4/3)x.

Another Example

This example uses the focal definition (difference of distances = 2a) with a vertical transverse axis, contrasting the first example which started from an equation with a horizontal transverse axis.

Problem: A hyperbola has foci at (0, −6) and (0, 6), and the constant difference of distances from any point on the hyperbola to the foci is 8. Write its equation in standard form.

Step 1: Since the foci are on the y-axis, the transverse axis is vertical. The standard form is y²/a² − x²/b² = 1. The center is at the origin.

\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1

Step 2: The constant difference of distances equals 2a. Set 2a = 8 and solve for a.

2a = 8 \implies a = 4,\quad a^2 = 16

Step 3: The distance from the center to each focus is c = 6, so c² = 36. Use b² = c² − a² to find b².

b^2 = 36 - 16 = 20

Step 4: Substitute the values into the vertical-transverse-axis standard form.

\frac{y^2}{16} - \frac{x^2}{20} = 1

Answer: The equation is y²/16 − x²/20 = 1.

Frequently Asked Questions

What is the difference between a hyperbola and an ellipse?

An ellipse is defined by the set of points where the sum of distances to two foci is constant, while a hyperbola uses the absolute difference of distances. This causes the ellipse to form one closed curve, whereas a hyperbola produces two open branches. Their standard-form equations also differ: an ellipse has a plus sign between the two fraction terms, and a hyperbola has a minus sign.

How do you find the asymptotes of a hyperbola?

For a hyperbola centered at the origin with horizontal transverse axis, x²/a² − y²/b² = 1, the asymptotes are y = ±(b/a)x. For a vertical transverse axis, y²/a² − x²/b² = 1, the asymptotes are y = ±(a/b)x. The branches of the hyperbola approach these lines but never touch them.

Why does a hyperbola have two branches?

The defining condition — that the absolute difference of distances to the two foci equals a constant — can be satisfied on two sides. A point can be closer to the left focus or closer to the right focus while still meeting the condition. Each possibility generates one branch, so the hyperbola always has exactly two separate, mirror-image curves.

Hyperbola vs. Ellipse

	Hyperbola	Ellipse
Defining property	Absolute difference of distances to foci is constant	Sum of distances to foci is constant
Standard form (horizontal)	x²/a² − y²/b² = 1	x²/a² + y²/b² = 1
Shape	Two open branches	One closed oval
Relationship of a, b, c	c² = a² + b² (c > a)	c² = a² − b² (c < a)
Asymptotes	Has two asymptotes	No asymptotes
Eccentricity	e > 1	0 < e < 1

Why It Matters

Hyperbolas appear frequently in precalculus and analytic geometry courses, often alongside the other conic sections on exams. They model real-world phenomena including the paths of certain comets, the shape of cooling tower structures, and the basis of GPS and LORAN navigation systems (which locate positions using differences of distances). Understanding hyperbolas is also essential for later work in multivariable calculus and physics, where hyperbolic orbits and inverse-square-law trajectories arise.

Common Mistakes

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

Correction: For a hyperbola, the foci are farther from the center than the vertices, so c > a. Always use c² = a² + b² when working with hyperbolas.

Mistake: Confusing which axis is the transverse axis based on the equation's form.

Correction: The transverse axis corresponds to whichever variable has the positive term. In x²/a² − y²/b² = 1, the transverse axis is horizontal (along x). In y²/a² − x²/b² = 1, it is vertical (along y). The positive term always comes first.

Related Terms

Conic Sections — Hyperbola is one of the four conic sections
Ellipse — Uses sum of distances instead of difference
Foci of a Hyperbola — The two fixed points defining the hyperbola
Directrices of a Hyperbola — Lines used in the focus-directrix definition
Focus (conic section) — General concept of a focal point for conics
Focal Radius — Distance from a point on the curve to a focus
Locus — Set of points satisfying a geometric condition
Point — Fundamental element making up the hyperbola