Hyperbola Formula

Hyperbola is the set of all points in a plane where the absolute difference of the distances to two fixed points (foci) is constant.

The Formula

Horizontal: (x−h)2a2−(y−k)2b2=1\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1
Vertical: (y−k)2a2−(x−h)2b2=1\frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1
Foci: c2=a2+b2c^2 = a^2 + b^2. Asymptotes: y−k=±ba(x−h)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

x216−y29=1\frac{x^2}{16} - \frac{y^2}{9} = 1 opens left-right. Vertices at (±4,0)(\pm 4, 0). Asymptotes: y=±34xy = \pm\frac{3}{4}x. Foci at (±5,0)(\pm 5, 0) since c=16+9=5c = \sqrt{16 + 9} = 5.

Notation

aa = distance from center to vertex, bb = used for asymptote slope, cc = 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)∣∣d((x,y),F1)−d((x,y),F2)∣=2a}\{(x,y) \mid |d((x,y), F_1) - d((x,y), F_2)| = 2a\}; standard form (x−h)2a2−(y−k)2b2=1\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1 with c2=a2+b2c^2 = a^2 + b^2, eccentricity e=ca>1e = \frac{c}{a} > 1

Worked Examples

Example 1

easy
Identify the vertices and the direction of opening for the hyperbola x29−y216=1\frac{x^2}{9} - \frac{y^2}{16} = 1.

Answer

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

First step

1
The standard form x2a2−y2b2=1\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 opens left and right (horizontally).

Full solution

  1. 2
    a2=9a^2 = 9, so a=3a = 3. The vertices are at (±a,0)=(±3,0)(\pm a, 0) = (\pm 3, 0).
  2. 3
    The transverse axis is along the xx-axis with vertices at (−3,0)(-3, 0) and (3,0)(3, 0).
In the standard form of a hyperbola, the positive fraction determines the direction of opening. When x2x^2 is positive, the hyperbola opens horizontally; when y2y^2 is positive, it opens vertically. The vertices are at distance aa from the center along the transverse axis.

Example 2

medium
Find the equations of the asymptotes for the hyperbola y24−x29=1\frac{y^2}{4} - \frac{x^2}{9} = 1.

Example 3

medium
Write the equation of a vertical hyperbola with vertices (0,±4)(0, \pm 4) and foci (0,±5)(0, \pm 5).

Common Mistakes

  • Using a2−b2a^2-b^2 for the foci - a hyperbola adds: c2=a2+b2c^2=a^2+b^2.
  • Reading the opening direction wrong - the branch opens along the variable of the POSITIVE term.
  • Mistaking the larger denominator for a2a^2 - in a hyperbola a2a^2 sits under the positive term regardless of size.

Why This Formula Matters

Hyperbolas model navigation (LORAN), comet paths, and shadow boundaries; the difference-of-distances property and the asymptote slopes are the defining skills. The c2=a2+b2c^2=a^2+b^2 relation (a PLUS, opposite the ellipse) and 'which variable comes first sets the opening direction' are the two facts students invert most. Recognizing it by "Is one squared term subtracted from the other (opposite signs) with the result equaling 1?" — rather than by familiar numbers — is what lets a student tell it apart from ellipse and asymptote and focus relation mix-up in a mixed problem set.

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.

How do you use the Hyperbola formula?

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.

What do the symbols mean in the Hyperbola formula?

aa = distance from center to vertex, bb = used for asymptote slope, cc = distance from center to focus. The transverse axis connects the vertices.

Why is the Hyperbola formula important in Math?

Hyperbolas model navigation (LORAN), comet paths, and shadow boundaries; the difference-of-distances property and the asymptote slopes are the defining skills. The c2=a2+b2c^2=a^2+b^2 relation (a PLUS, opposite the ellipse) and 'which variable comes first sets the opening direction' are the two facts students invert most. Recognizing it by "Is one squared term subtracted from the other (opposite signs) with the result equaling 1?" — rather than by familiar numbers — is what lets a student tell it apart from ellipse and asymptote and focus relation mix-up in a mixed problem set.

What do students get wrong about Hyperbola?

The procedure for hyperbola is the easy part; the trap is using a2−b2a^2-b^2 for the foci. Asking "Is one squared term subtracted from the other (opposite signs) with the result equaling 1?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

What should I learn before the Hyperbola formula?

Before studying the Hyperbola formula, you should understand: equation of circle, asymptote.

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