Hyperbola Formula

Q: What do the symbols mean in the Hyperbola formula?

$a$ = distance from center to vertex, $b$ = used for asymptote slope, $c$ = distance from center to focus. The transverse axis connects the vertices.

The Formula

Horizontal: \frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1
Vertical: \frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1
Foci: c^2 = a^2 + b^2. Asymptotes: y - k = \pm\frac{b}{a}(x - h) (horizontal opening).

When to use: While an ellipse keeps the SUM of distances to foci constant, a hyperbola keeps the DIFFERENCE constant. This creates two separate curves that open away from each other, each curving toward (but never reaching) a pair of asymptotic lines.

Quick Example

\frac{x^2}{16} - \frac{y^2}{9} = 1 opens left-right. Vertices at (\pm 4, 0). Asymptotes: y = \pm\frac{3}{4}x. Foci at (\pm 5, 0) since c = \sqrt{16 + 9} = 5.

Notation

a = distance from center to vertex, b = used for asymptote slope, c = distance from center to focus. The transverse axis connects the vertices.

What This Formula Means

The set of all points in a plane where the absolute difference of the distances to two fixed points (foci) is constant. The curve has two separate branches and asymptotes.

While an ellipse keeps the SUM of distances to foci constant, a hyperbola keeps the DIFFERENCE constant. This creates two separate curves that open away from each other, each curving toward (but never reaching) a pair of asymptotic lines.

Formal View

\{(x,y) \mid |d((x,y), F_1) - d((x,y), F_2)| = 2a\}; standard form \frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1 with c^2 = a^2 + b^2, eccentricity e = \frac{c}{a} > 1

Worked Examples

Example 1

easy

Identify the vertices and the direction of opening for the hyperbola \frac{x^2}{9} - \frac{y^2}{16} = 1.

Solution

1
The standard form \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 opens left and right (horizontally).
2
a^2 = 9, so a = 3. The vertices are at (\pm a, 0) = (\pm 3, 0).
3
The transverse axis is along the x-axis with vertices at (-3, 0) and (3, 0).

Answer

\text{Vertices: } (\pm 3, 0); \text{ opens left and right}

In the standard form of a hyperbola, the positive fraction determines the direction of opening. When x^2 is positive, the hyperbola opens horizontally; when y^2 is positive, it opens vertically. The vertices are at distance a from the center along the transverse axis.

Example 2

medium

Find the equations of the asymptotes for the hyperbola \frac{y^2}{4} - \frac{x^2}{9} = 1.

Common Mistakes

Using c^2 = a^2 - b^2 (the ellipse formula) instead of c^2 = a^2 + b^2 for hyperbolas. Remember: for hyperbolas, c > a.
Confusing opening direction: the variable with the POSITIVE sign determines the opening. \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 opens left-right; \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 opens up-down.
Getting asymptote slopes backwards: for \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1, the asymptotes are y = \pm\frac{b}{a}x, NOT \pm\frac{a}{b}x.

Why This Formula Matters

Hyperbolas model sonic booms, orbits of comets that don't return, navigation systems (LORAN), and the relationship between pressure and volume. Hyperbolic shapes appear in cooling towers and telescope mirrors.

Frequently Asked Questions

What is the Hyperbola formula?

The set of all points in a plane where the absolute difference of the distances to two fixed points (foci) is constant. The curve has two separate branches and asymptotes.