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Hyperbola Calculator

Enter a hyperbola in standard or general form — or from its center, a vertex and a focus — to get every property, a graph and the steps.

(x − 2)²/9 − (y + 1)²/16 = 1

Orientation
Horizontal (opens left/right)
Center
(2, -1)
a, b, c
3, 4, 5
Vertices
(-1, -1), (5, -1)
Co-vertices
(2, -5), (2, 3)
Foci
(-3, -1), (7, -1)
Asymptote 1
y = 1.3333x − 3.6667
Asymptote 2
y = -1.3333x + 1.6667
Eccentricity e
1.6667
Transverse axis (2a)
6
Conjugate axis (2b)
8
Latus rectum (2b²/a)
10.6667
  1. a = √9 = 3, b = √16 = 4.
  2. c = √(a² + b²) = √25 = 5.
  3. e = c / a = 1.6667.
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Hyperbola vs ellipse: why c² = a² + b²

An ellipse is the set of points whose distances to two foci add to a constant; a hyperbola is the set whose distances differ by a constant (2a). That swap from sum to difference flips the sign in the equation and moves the foci outside the vertices, so c > a and c² = a² + b² — the ellipse relation c² = a² − b² does not apply.

Horizontal vs vertical hyperbolas

a² always sits under the positive term. If x² is positive the branches open left and right and the asymptote slopes are ±b/a; if y² is positive they open up and down and the slopes are ±a/b.

CV1V2F1F2
x²/4 − y²/2.25 = 1 (horizontal)
CV1V2F1F2
y²/4 − x²/2.25 = 1 (vertical)

Worked examples

1. (x − 2)²/9 − (y + 1)²/16 = 1

a = 3, b = 4, c = 5: center (2, -1), vertices (-1, -1) and (5, -1), foci (-3, -1) and (7, -1), asymptotes y = 1.3333x − 3.6667 and y = -1.3333x + 1.6667, e = 1.6667.

2. y²/25 − x²/144 = 1

Vertical; a = 5, b = 12, c = 13: center (0, 0), vertices (0, -5) and (0, 5), foci (0, -13) and (0, 13), asymptotes y = 0.4167x and y = -0.4167x, e = 2.6.

3. 9x² − 4y² − 36x − 8y − 4 = 0 to standard form

  1. Group terms: 9(x² − 4x) − 4(y² + 2y) = 4.
  2. Complete the square: add 9·4 = 36 and -4·1 = -4 to both sides.
  3. 9(x − 2)² − 4(y + 1)² = 36.
  4. Divide by 36: (x − 2)²/4 − (y + 1)²/9 = 1.
  5. The positive term is the x term, so the hyperbola opens left and right; a² = 4, b² = 9.

center (2, -1), vertices (0, -1) and (4, -1), foci (-1.6056, -1) and (5.6056, -1), asymptotes y = 1.5x − 4 and y = -1.5x + 2, e = 1.8028.

Common mistakes

  • Putting a under the wrong term — a² is always under the positive term, even when it is the smaller number.
  • Using c² = a² − b² — that is the ellipse formula; for a hyperbola c² = a² + b².
  • Asymptote slope for vertical hyperbolas — it is ±a/b, not ±b/a.
  • Sign errors in the center — (x + 3)² means h = −3, not +3.

Frequently asked questions

How do you find the foci of a hyperbola?

Write the equation in standard form, read a² (under the positive term) and b², then compute c = √(a² + b²). The foci sit c units from the center along the transverse axis: (h ± c, k) for a horizontal hyperbola, (h, k ± c) for a vertical one.

How do you find the asymptotes of a hyperbola?

For a horizontal hyperbola the asymptotes are y − k = ±(b/a)(x − h). For a vertical hyperbola they are y − k = ±(a/b)(x − h). Both pass through the center.

How do you find the eccentricity of a hyperbola?

Eccentricity is e = c/a, where c = √(a² + b²). Because c is always larger than a, a hyperbola's eccentricity is always greater than 1.

How can you tell a hyperbola equation from an ellipse equation?

In standard form a hyperbola has a minus sign between the squared terms and an ellipse has a plus sign. In general form Ax² + Cy² + Dx + Ey + F = 0, A and C have opposite signs for a hyperbola and the same sign for an ellipse.

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