Hyperbola Properties Calculator

Introduction: Finding Standard Hyperbola Properties

This hyperbola properties calculator evaluates a standard, axis-aligned hyperbola from its center (h, k), positive semi-axis lengths a and b, and horizontal or vertical orientation. It returns the standard-form equation, vertices, foci, asymptotes, the focal distance c, and eccentricity. The discussion below connects each displayed value to the geometry of a hyperbola used in algebra and analytic geometry.

Standard Forms for the Entered Hyperbola

A hyperbola has two separate branches and is defined by a constant difference between the distances from a point on the curve to two fixed points called foci. This calculator uses the usual centered standard forms, in which (h, k) is the center of symmetry.

The selected orientation determines which squared coordinate term is positive:

• Horizontal transverse axis (opens left–right):
• Vertical transverse axis (opens up–down):

Enter values that match one of these forms: a belongs below the positive term and measures along the opening direction, while b belongs below the subtracted term. The calculator then inserts the entered numbers into the matching standard-form equation.

Hyperbola Parameters a, b, and c

For the hyperbola entered here, a and b control different center-to-axis distances:

• a is the distance from the center to either vertex along the transverse axis, the direction in which the branches open.
• b is the distance from the center along the perpendicular conjugate axis; it helps determine the slopes of the asymptotes.

The focal distance c is the distance from the center to either focus. For every nondegenerate standard hyperbola, these parameters satisfy

Thus the calculator obtains c as the positive square root of a² + b². Unlike an ellipse, a hyperbola has c > a, so its foci lie farther from the center than its vertices.

Hyperbola Vertices, Co-vertices, and Foci

The center coordinates supplied to the hyperbola calculator locate every special point, while the orientation decides whether the transverse axis is horizontal or vertical.

Horizontal hyperbola point locations

• Vertices: (h ± a, k)
• Co-vertices: (h, k ± b)
• Foci: (h ± c, k), where c = √(a² + b²)

Vertical hyperbola point locations

• Vertices: (h, k ± a)
• Co-vertices: (h ± b, k)
• Foci: (h, k ± c), where c = √(a² + b²)

The results table reports numerical coordinates for the vertices and foci. Co-vertices are useful construction points for sketching the guiding rectangle, but they are not separately listed in the calculator output; apply the formulas above when you need them.

Hyperbola Asymptotes and Eccentricity

A standard hyperbola’s asymptotes pass through its center and give the limiting directions of the two branches as they extend away from that center.

Asymptotes for each hyperbola orientation

• Horizontal hyperbola (transverse axis along the x-direction):
  • y - k = (b/a)(x - h)
  • y - k = -(b/a)(x - h)
• Vertical hyperbola (transverse axis along the y-direction):
  • y - k = (a/b)(x - h)
  • y - k = -(a/b)(x - h)

The calculator displays this pair compactly with a ± sign and substitutes the center and the appropriate slope ratio. Switching orientation changes the asymptote slope from b/a to a/b.

Hyperbola eccentricity

The hyperbola’s eccentricity compares its focal distance with its transverse semi-axis length:

Because c² = a² + b² and both semi-axis lengths must be positive, c > a and therefore e > 1. Increasing b while holding a fixed increases both c and eccentricity; changing a also changes the vertex distance and the asymptote slopes.

How to Use the Hyperbola Properties Calculator

To calculate properties of a standard hyperbola, enter the values that identify its center, scale, and opening direction.

1. Enter the center: type the coordinates h and k of the center. For a hyperbola centered at the origin, enter h = 0 and k = 0.
2. Enter positive semi-axis lengths: provide positive real values for a and b. Use a for the semi-axis in the direction the hyperbola opens and b for the perpendicular semi-axis.
3. Choose orientation: select Horizontal when the positive term in the standard equation contains x; select Vertical when the positive term contains y.
4. Click calculate: the tool produces the substituted equation, vertex coordinates, focus coordinates, c, eccentricity, and the pair of asymptotes.

Check the sign and placement of the squared terms before calculating. A correct center and semi-axis lengths paired with the wrong orientation describe a different hyperbola.

Interpreting Hyperbola Properties Results

The hyperbola results table translates the entered standard-form parameters into the following graphing information:

• Equation: the calculator’s numerical standard form using the selected orientation and the squared values a² and b².
• Vertices: the nearest points on the branches to the center, located a units from the center on the transverse axis.
• Foci: the defining fixed points, located c units from the center on the same transverse axis.
• c: the focal distance √(a² + b²).
• Eccentricity: the ratio c/a, which is always greater than 1 for the valid inputs used by this calculator.
• Asymptotes: the two center-crossing lines that the branches approach without meeting.

To sketch the calculated hyperbola, plot its center and vertices first. Mark points b units from the center on the conjugate axis to form a rectangle of width 2a and height 2b, with the dimensions exchanged visually for the vertical case. Its diagonals provide the asymptote directions; then draw the branches opening through the vertices and approaching those diagonals.

Worked Example: Horizontal Hyperbola with a = 3 and b = 2

For a concrete horizontal-hyperbola calculation, use the calculator’s displayed starting values:

• h = 0
• k = 0
• a = 3
• b = 2
• Orientation: Horizontal

The corresponding standard-form hyperbola is

x²/9 - y²/4 = 1.

Applying the same relationships used by the calculator gives:

• Center: (h, k) = (0, 0).
• Vertices: (h ± a, k) = (± 3, 0).
• Co-vertices for graph construction: (h, k ± b) = (0, ± 2).
• c = √(a² + b²) = √(3² + 2²) = √(9 + 4) = √13, so the foci are (h ± c, k) = (± √13, 0).
• The horizontal asymptotes are y = (b/a)x and y = -(b/a)x. Since b/a = 2/3, they become y = (2/3)x and y = -(2/3)x.
• Eccentricity: e = c/a = √13 / 3, which is greater than 1.

On submission, the calculator expresses the focus coordinates, c, eccentricity, and asymptote slope as decimal values rounded to three places. The equation and vertex coordinates are likewise built from the values entered in the form.

Comparison: Horizontal and Vertical Hyperbola Properties

This comparison shows exactly what changes in the calculator’s hyperbola results when the same h, k, a, and b are assigned the other orientation.

Standard Hyperbola Assumptions, Limitations, and Notes

These limits describe the class of hyperbolas handled by this properties calculator.

• The calculator accepts a hyperbola already in standard form with center (h, k); it does not convert a general second-degree equation into that form.
• The parameters a and b must be positive real numbers. Zero or negative entries do not provide the usual geometric semi-axis lengths and can lead to invalid or nonmeaningful results.
• The selected orientation determines whether the transverse axis is horizontal or vertical. Select the orientation from the positive squared term in the original equation, not from the relative sizes of a and b.
• The calculator rounds its displayed numerical values to three decimal places. Exact radical forms, such as √13, are represented numerically in the results table.
• Very large or very small values of a or b may make rounding more apparent in the displayed coordinates, focal distance, eccentricity, and asymptote slope.
• This tool calculates geometric properties only. It does not graph the curve or process rotated hyperbolas whose axes are not parallel to the coordinate axes.

With those conditions satisfied, the center coordinates, positive semi-axis lengths, and orientation fully determine the standard hyperbola properties reported here.

Formula: Hyperbola Property Calculations

For the entered standard hyperbola, the calculator first finds the focal distance and eccentricity from c = √(a² + b²) and e = c/a. It then uses h, k, and the selected orientation to place the vertices and foci and to choose an asymptote slope of b/a for a horizontal hyperbola or a/b for a vertical hyperbola. Enter h and k as coordinate values and a and b as positive lengths in the same coordinate scale.