Online Hyperbola Calculator
A hyperbola is one of the four fundamental conic sections with unique mathematical properties essential to algebra, geometry, calculus, and real-world applications. This comprehensive online hyperbola calculator helps students, educators, and mathematicians calculate all key properties including center, vertices, co-vertices, foci, asymptotes, eccentricity, transverse and conjugate axes with step-by-step solutions and properly formatted mathematical notation.
Select Hyperbola Orientation
Standard Form Equation:
\[\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1\]
Center Coordinates
Semi-Axes Lengths
Understanding Hyperbolas
A hyperbola is a conic section formed when a plane intersects both nappes (cones) of a double cone. Unlike ellipses which are closed curves, hyperbolas consist of two separate branches that mirror each other. Mathematically, a hyperbola is defined as the set of all points where the absolute difference of distances from two fixed points (called foci) remains constant. This fundamental property distinguishes hyperbolas from other conic sections and gives rise to their distinctive shape and mathematical characteristics.
Standard Form Equations of Hyperbolas
Hyperbola Equation Forms:
Horizontal Hyperbola (opens left and right):
\[\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1\]
Center: \((h, k)\), Transverse axis: horizontal
Vertical Hyperbola (opens up and down):
\[\frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1\]
Center: \((h, k)\), Transverse axis: vertical
Key Identification Rule: The positive term determines orientation. If \(x^2\) term is positive → horizontal. If \(y^2\) term is positive → vertical.
Parameters:
\(a\) = distance from center to vertex along transverse axis
\(b\) = distance from center to co-vertex along conjugate axis
\(c\) = distance from center to focus, where \(c^2 = a^2 + b^2\)
Key Hyperbola Properties and Formulas
Essential Hyperbola Formulas:
Relationship Between a, b, and c:
\[c^2 = a^2 + b^2\]
Note: For hyperbolas, \(c > a\) always (unlike ellipses where \(c < a\))
Eccentricity:
\[e = \frac{c}{a}\]
For all hyperbolas: \(e > 1\)
Vertices (Horizontal Hyperbola):
\[V_1 = (h-a, k), \quad V_2 = (h+a, k)\]
Vertices (Vertical Hyperbola):
\[V_1 = (h, k-a), \quad V_2 = (h, k+a)\]
Foci (Horizontal Hyperbola):
\[F_1 = (h-c, k), \quad F_2 = (h+c, k)\]
Foci (Vertical Hyperbola):
\[F_1 = (h, k-c), \quad F_2 = (h, k+c)\]
Asymptotes (Horizontal Hyperbola):
\[y - k = \pm\frac{b}{a}(x - h)\]
Asymptotes (Vertical Hyperbola):
\[y - k = \pm\frac{a}{b}(x - h)\]
Comprehensive Hyperbola Example
Example: Find All Properties of the Hyperbola
Given Equation: \[\frac{(x-2)^2}{9} - \frac{(y+1)^2}{16} = 1\]
Step 1: Identify Orientation and Center
Since the \(x^2\) term is positive, this is a horizontal hyperbola
Center: \((h, k) = (2, -1)\)
Step 2: Determine a and b
\(a^2 = 9\), so \(a = 3\)
\(b^2 = 16\), so \(b = 4\)
Step 3: Calculate c
\(c^2 = a^2 + b^2 = 9 + 16 = 25\)
\(c = 5\)
Step 4: Find Vertices (horizontal, so along x-axis from center)
\(V_1 = (h-a, k) = (2-3, -1) = (-1, -1)\)
\(V_2 = (h+a, k) = (2+3, -1) = (5, -1)\)
Step 5: Find Co-vertices
\(CV_1 = (h, k-b) = (2, -1-4) = (2, -5)\)
\(CV_2 = (h, k+b) = (2, -1+4) = (2, 3)\)
Step 6: Find Foci
\(F_1 = (h-c, k) = (2-5, -1) = (-3, -1)\)
\(F_2 = (h+c, k) = (2+5, -1) = (7, -1)\)
Step 7: Calculate Eccentricity
\(e = \frac{c}{a} = \frac{5}{3} \approx 1.667\)
Step 8: Find Asymptotes
\(y - (-1) = \pm\frac{4}{3}(x - 2)\)
\(y + 1 = \pm\frac{4}{3}(x - 2)\)
Asymptote 1: \(y = \frac{4}{3}x - \frac{11}{3}\)
Asymptote 2: \(y = -\frac{4}{3}x + \frac{5}{3}\)
Components of a Hyperbola
Understanding each component of a hyperbola enables accurate graphing, problem-solving, and application to real-world scenarios. Each element has geometric significance and mathematical relationships with other components.
| Component | Definition | Horizontal Hyperbola | Vertical Hyperbola |
|---|---|---|---|
| Center | Midpoint between vertices | (h, k) | (h, k) |
| Vertices | Points on transverse axis | (h±a, k) | (h, k±a) |
| Co-vertices | Points on conjugate axis | (h, k±b) | (h±b, k) |
| Foci | Fixed points, c² = a² + b² | (h±c, k) | (h, k±c) |
| Transverse Axis | Axis through vertices | Horizontal, length 2a | Vertical, length 2a |
| Conjugate Axis | Axis perpendicular to transverse | Vertical, length 2b | Horizontal, length 2b |
| Asymptotes | Lines hyperbola approaches | y-k = ±(b/a)(x-h) | y-k = ±(a/b)(x-h) |
Hyperbola vs. Other Conic Sections
Hyperbolas are one of four conic sections, each with distinct properties and equations. Understanding differences helps identify which conic section an equation represents and apply appropriate solution techniques.
| Conic Section | Eccentricity (e) | General Form | Key Characteristic |
|---|---|---|---|
| Circle | e = 0 | \((x-h)^2 + (y-k)^2 = r^2\) | All points equidistant from center |
| Ellipse | 0 < e < 1 | \(\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\) | Sum of distances to foci constant |
| Parabola | e = 1 | \((y-k)^2 = 4p(x-h)\) | Equidistant from focus and directrix |
| Hyperbola | e > 1 | \(\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1\) | Difference of distances to foci constant |
Equation Identification Tip: In standard form, if both squared terms have the same sign (both positive) → ellipse or circle. If squared terms have opposite signs (one positive, one negative) → hyperbola. If only one variable is squared → parabola. This quick check helps immediately classify conic sections when analyzing equations.
Eccentricity and Hyperbola Shape
Eccentricity measures how "stretched" or "open" a hyperbola is. For all hyperbolas, eccentricity is always greater than 1, distinguishing them from ellipses and parabolas. Understanding eccentricity helps visualize hyperbola shape and predict behavior.
Eccentricity Analysis:
Definition: \(e = \frac{c}{a}\) where \(c = \sqrt{a^2 + b^2}\)
Alternative Form: \(e = \sqrt{1 + \frac{b^2}{a^2}}\)
Eccentricity Range and Shape:
- \(e\) slightly greater than 1 (e.g., 1.1-1.3): Hyperbola is relatively "closed" with narrow opening
- \(e\) moderate (e.g., 1.5-2.0): Standard hyperbola shape with moderate opening
- \(e\) large (e.g., 3.0+): Hyperbola is very "open" with branches nearly parallel to asymptotes
Relationship to Asymptotes: As eccentricity increases, hyperbola branches approach their asymptotes more quickly and spread wider apart.
Converting General Form to Standard Form
Hyperbola equations often appear in general form requiring algebraic manipulation to convert to standard form for analysis. Completing the square is the primary technique for this conversion.
Example: Convert General Form to Standard Form
Given: \(9x^2 - 4y^2 - 36x - 24y - 36 = 0\)
Step 1: Rearrange by grouping like terms
\(9x^2 - 36x - 4y^2 - 24y = 36\)
Step 2: Factor coefficients
\(9(x^2 - 4x) - 4(y^2 + 6y) = 36\)
Step 3: Complete the square for x
\(x^2 - 4x\): take half of -4, square it: \((-2)^2 = 4\)
\(9(x^2 - 4x + 4 - 4) = 9((x-2)^2 - 4) = 9(x-2)^2 - 36\)
Step 4: Complete the square for y
\(y^2 + 6y\): take half of 6, square it: \((3)^2 = 9\)
\(-4(y^2 + 6y + 9 - 9) = -4((y+3)^2 - 9) = -4(y+3)^2 + 36\)
Step 5: Substitute back
\(9(x-2)^2 - 36 - 4(y+3)^2 + 36 = 36\)
\(9(x-2)^2 - 4(y+3)^2 = 36\)
Step 6: Divide by 36 to get standard form
\[\frac{(x-2)^2}{4} - \frac{(y+3)^2}{9} = 1\]
Result: Center (2, -3), horizontal hyperbola, \(a^2 = 4\) so \(a = 2\), \(b^2 = 9\) so \(b = 3\)
Real-World Applications of Hyperbolas
Hyperbolas appear in numerous practical applications across physics, engineering, astronomy, and technology. Understanding these applications demonstrates the relevance of hyperbolic mathematics beyond theoretical study.
Common Hyperbola Applications
- Navigation Systems (LORAN): Long Range Navigation uses hyperbolic curves to determine position. Ships and aircraft measure time differences of radio signals from two stations, placing them on a hyperbola with those stations as foci.
- Sonic Booms: When objects travel faster than sound, shock waves form hyperbolic patterns. The hyperbola's shape depends on the object's speed relative to sound speed (Mach number).
- Orbital Mechanics: Objects with sufficient velocity escape gravitational pull following hyperbolic trajectories. Comets and spacecraft slingshot maneuvers often follow hyperbolic paths around planets or stars.
- Telescope Mirrors: Hyperbolic mirrors reflect light to a single focal point. Combined with parabolic mirrors in Cassegrain telescopes, they provide clear images with compact designs.
- Cooling Towers: Nuclear and coal power plant cooling towers use hyperbolic shapes for structural strength and optimal air circulation. The hyperbola provides maximum strength with minimum material.
- Particle Physics: In particle accelerators, hyperbolic paths describe particle trajectories under certain force conditions. Charged particles in magnetic fields may follow hyperbolic curves.
- Acoustics: In architectural acoustics, hyperbolic reflectors focus sound waves to specific points, useful in concert halls and auditoriums for sound distribution.
Graphing Hyperbolas Step-by-Step
Accurate hyperbola graphing requires systematic approach plotting key points and understanding asymptotic behavior. Following these steps ensures correct representation of hyperbolic curves.
Graphing Procedure:
Step 1: Identify and plot the center (h, k)
Step 2: Determine orientation (horizontal or vertical from equation)
Step 3: Plot vertices at distance a from center along transverse axis
Step 4: Plot co-vertices at distance b from center along conjugate axis
Step 5: Create fundamental rectangle with vertices and co-vertices as corners
Step 6: Draw asymptotes through center along rectangle diagonals
Step 7: Plot foci at distance c from center along transverse axis
Step 8: Sketch branches starting at vertices, approaching asymptotes, passing through foci region
Remember: Hyperbola branches never cross asymptotes, they approach infinitely close but never touch. Branches always "bend away" from conjugate axis, opening toward transverse axis direction.
Special Cases and Degenerate Hyperbolas
Certain coefficient relationships create special or degenerate cases of hyperbolas with unique properties or non-standard forms requiring special consideration.
Rectangular (Equilateral) Hyperbola
Definition: When \(a = b\), the hyperbola is called rectangular or equilateral. Its asymptotes are perpendicular (slopes are negative reciprocals), forming 90° angles. The eccentricity of rectangular hyperbolas is always \(e = \sqrt{2} \approx 1.414\). The standard form \(xy = k\) represents a rectangular hyperbola rotated 45° with asymptotes along the coordinate axes.
Degenerate Cases
- Two Intersecting Lines: When right side equals 0: \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 0\) factors to \(\left(\frac{x}{a} - \frac{y}{b}\right)\left(\frac{x}{a} + \frac{y}{b}\right) = 0\), giving two straight lines through the origin (the asymptotes themselves).
- No Graph: If right side is negative: \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = -1\), no real points satisfy the equation, producing no graph (though imaginary solutions exist).
- Single Point: In limiting cases with specific coefficient relationships, the hyperbola can degenerate to a single point.
Common Mistakes When Working with Hyperbolas
- Confusing a and b roles: Unlike ellipses where a is always larger, in hyperbolas a and b don't have size relationship. The value a is always under the positive term in standard form, regardless of magnitude.
- Using wrong c formula: For hyperbolas, \(c^2 = a^2 + b^2\) (addition). For ellipses, \(c^2 = a^2 - b^2\) (subtraction). Mixing these creates incorrect foci locations.
- Misidentifying orientation: Always check which squared term is positive. Positive x² term means horizontal, positive y² term means vertical, regardless of a and b values.
- Incorrect asymptote slopes: For horizontal hyperbolas, slope is ±b/a. For vertical hyperbolas, slope is ±a/b. These are different formulas that cannot be interchanged.
- Forgetting center translation: When center is not at origin, every point (vertices, foci, co-vertices) must account for (h, k) shift. Forgetting translation creates graphs in wrong location.
- Assuming foci between vertices: Hyperbola foci lie outside the vertices (c > a). If foci appear between vertices, calculation error occurred.
Frequently Asked Questions
What is a hyperbola in mathematics?
A hyperbola is a type of conic section formed by intersecting a double cone with a plane at an angle that produces two separate curves called branches. Mathematically, a hyperbola is the set of all points where the absolute difference of distances from two fixed points (foci) is constant. The general equation is \(\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1\) for horizontal or \(\frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1\) for vertical hyperbolas. Hyperbolas have two branches, two vertices, two foci, and two asymptotes that the curves approach infinitely but never touch.
What is the standard form equation of a hyperbola?
The standard form of a horizontal hyperbola centered at (h, k) is: \(\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1\). For a vertical hyperbola: \(\frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1\). The positive term determines orientation: if x² term is positive, hyperbola opens left/right (horizontal); if y² term is positive, it opens up/down (vertical). Parameters a and b determine shape, with vertices at distance a from center along the transverse axis. The relationship \(c^2 = a^2 + b^2\) determines foci location.
How do you find the foci of a hyperbola?
Find the foci using the relationship \(c^2 = a^2 + b^2\), where c is the distance from center to each focus. For a horizontal hyperbola centered at (h, k), foci are at (h+c, k) and (h-c, k). For a vertical hyperbola, foci are at (h, k+c) and (h, k-c). The foci always lie on the transverse axis beyond the vertices. Note that for hyperbolas, c is always greater than a (unlike ellipses where c < a), so foci lie outside the vertices, not between them.
What are the asymptotes of a hyperbola?
Asymptotes are diagonal lines that the hyperbola branches approach infinitely but never touch or cross. For a horizontal hyperbola \(\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1\), asymptotes are: \(y - k = \pm\frac{b}{a}(x - h)\). For a vertical hyperbola \(\frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1\), asymptotes are: \(y - k = \pm\frac{a}{b}(x - h)\). These lines pass through the center (h, k) and form diagonals of the fundamental rectangle created by points (h±a, k±b), providing visual guides for sketching the hyperbola.
What is the eccentricity of a hyperbola?
Eccentricity (e) measures how "open" or stretched a hyperbola is, calculated as \(e = \frac{c}{a}\) where c is the distance to foci and a is the distance to vertices. For all hyperbolas, e > 1 (always greater than one). Values close to 1 (like 1.1-1.3) create relatively closed hyperbolas with narrow openings. Larger eccentricities (2, 3, or more) create wider, more open hyperbolas. Eccentricity distinguishes hyperbolas (e > 1) from ellipses (0 < e < 1), parabolas (e = 1), and circles (e = 0).
How do you know if an equation represents a hyperbola?
An equation represents a hyperbola if it has two squared variable terms with opposite signs (one positive, one negative) and equals a constant. Standard form: \(\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = \pm 1\). If both squared terms have the same sign → ellipse or circle. If only one variable is squared → parabola. The minus sign between squared terms is the key identifier for hyperbolas. In general form \(Ax^2 + Cy^2 + Dx + Ey + F = 0\), if A and C have opposite signs (one positive, one negative), the equation represents a hyperbola.
What is the difference between horizontal and vertical hyperbolas?
Horizontal hyperbolas open left and right, with equation \(\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1\) (positive x² term), vertices at (h±a, k), and transverse axis horizontal. Vertical hyperbolas open up and down, with equation \(\frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1\) (positive y² term), vertices at (h, k±a), and transverse axis vertical. The orientation is determined solely by which squared term is positive in standard form, not by the relative sizes of a and b. Asymptote slopes also differ: ±b/a for horizontal, ±a/b for vertical.
How do you graph a hyperbola?
To graph a hyperbola: 1) Plot center (h, k), 2) From standard form, identify a and b values, 3) Plot vertices at distance a from center along transverse axis (direction determined by positive term), 4) Plot co-vertices at distance b from center along conjugate axis, 5) Draw fundamental rectangle through these four points, 6) Draw asymptotes as diagonals through this rectangle, 7) Calculate c using \(c^2 = a^2 + b^2\) and plot foci at distance c from center along transverse axis, 8) Sketch hyperbola branches starting at vertices, passing near foci, approaching asymptotes. Branches never touch asymptotes or cross to other side.
