Hyperbola Calculator

Explore hyperbolas with vertices, foci, asymptotes, eccentricity, and equation conversion.

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Center h Center k Semi-transverse axis a Semi-conjugate axis b Opening

Conversion and properties

Equation Enter valid hyperbola data.

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What Is a Hyperbola?

A hyperbola is a conic section with two separate branches. One way to define it is as the set of all points in a plane where the absolute difference between the distances to two fixed points is constant. The two fixed points are called the foci.

In coordinate geometry, a hyperbola is usually studied through its equation. The equation tells you where the center is, which direction the branches open, where the vertices and foci are, and which straight lines the branches approach as they extend outward.

A hyperbola is different from an ellipse even though both use two foci. For an ellipse, the sum of the distances to the foci is constant. For a hyperbola, the difference of the distances is constant. That change in the distance rule is why a hyperbola has two open branches instead of one closed oval.

Why Hyperbolas Matter

Hyperbolas are important because they connect algebra, geometry, and graphing. A single equation can describe a curve's location, orientation, axes, foci, asymptotes, and eccentricity.

For students, hyperbolas are a key part of conic sections. Learning them helps with:

recognizing standard forms of conic equations;
completing the square to rewrite a general equation;
graphing from algebraic information;
distinguishing hyperbolas from ellipses and parabolas;
understanding how foci, vertices, and asymptotes shape a curve.

The most useful habit is to read the equation structurally. The signs, denominators, and center terms each tell you something specific about the graph.

Key Terms to Know

Center: The midpoint between the two foci. In standard form, the center is $(h,k)$.
Foci: The two fixed points used in the distance-difference definition of a hyperbola.
Focal distance: The distance from the center to each focus, usually written as $c$.
Transverse axis: The axis that passes through the center, vertices, and foci.
Conjugate axis: The axis through the center that is perpendicular to the transverse axis.
Vertex: A point where a branch meets the transverse axis. The distance from the center to each vertex is $a$.
Semi-transverse axis: The length $a$, which is half of the transverse axis length.
Semi-conjugate axis: The length $b$, which helps define the guide rectangle and asymptote slopes.
Eccentricity: The ratio $e=\frac{c}{a}$. For a valid hyperbola, $e>1$.
Asymptotes: Straight lines that the branches approach as they extend outward.
Standard form: A form of the equation that makes the center, orientation, and axis values easy to identify.

How Hyperbolas Work

The standard equation of a hyperbola depends on whether the transverse axis is horizontal or vertical.

A horizontal hyperbola has the form:

$$ \frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1 $$

It opens left and right. Its vertices are $(h-a,k)$ and $(h+a,k)$, and its foci are $(h-c,k)$ and $(h+c,k)$.

A vertical hyperbola has the form:

$$ \frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1 $$

It opens up and down. Its vertices are $(h,k-a)$ and $(h,k+a)$, and its foci are $(h,k-c)$ and $(h,k+c)$.

In both forms, the relationship among $a$, $b$, and $c$ is:

$$ c^2 = a^2 + b^2 $$

This is one of the most important formulas for a hyperbola. It also explains why the foci are farther from the center than the vertices: since $c^2=a^2+b^2$, $c$ must be greater than $a$ when $b>0$.

The eccentricity is:

$$ e = \frac{c}{a} $$

Because $c>a$, the eccentricity of a hyperbola is always greater than $1$.

How the Opening Direction Is Determined

The positive squared term determines the direction of the transverse axis.

If the $x$-term is positive, the hyperbola opens left and right:

$$ \frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1 $$

If the $y$-term is positive, the hyperbola opens up and down:

$$ \frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1 $$

This sign pattern is often the fastest way to avoid graphing the hyperbola in the wrong direction.

How Asymptotes Are Found

The asymptotes pass through the center and guide the direction of the two branches.

For a horizontal hyperbola:

$$ y-k = \pm \frac{b}{a}(x-h) $$

For a vertical hyperbola:

$$ y-k = \pm \frac{a}{b}(x-h) $$

The asymptotes are not part of the hyperbola. They are reference lines that the branches approach.

How General Form Becomes Standard Form

A supported axis-aligned general form can be written as:

$$ Ax^2 + Cy^2 + Dx + Ey + F = 0 $$

For this type of equation to represent an axis-aligned hyperbola, the $x^2$ and $y^2$ coefficients must have opposite signs. That is why one squared term becomes positive in standard form while the other becomes negative.

To rewrite a general equation in standard form:

Group the $x$ terms and $y$ terms.
Move the constant term to the other side.
Complete the square for $x$ and $y$ when needed.
Divide so the right side becomes $1$.
Read the center, $a$, $b$, orientation, vertices, foci, and asymptotes from the standard form.

For the supported form above, the center can be found from the coefficients:

$$ h = -\frac{D}{2A} $$

$$ k = -\frac{E}{2C} $$

This works for non-rotated hyperbolas that do not contain an $xy$ term.

Examples of Hyperbolas in Practice

Example 1: Reading a Horizontal Hyperbola from Standard Form

Suppose a hyperbola has center $(2,-1)$, semi-transverse axis $a=3$, semi-conjugate axis $b=4$, and horizontal orientation.

The standard equation is:

$$ \frac{(x-2)^2}{9} - \frac{(y+1)^2}{16} = 1 $$

The focal distance is:

$$ c = \sqrt{a^2+b^2}=\sqrt{3^2+4^2}=5 $$

The eccentricity is:

$$ e = \frac{c}{a}=\frac{5}{3}\approx 1.6667 $$

Because the hyperbola is horizontal, the vertices are:

$$ (2-3,-1)=(-1,-1) $$

$$ (2+3,-1)=(5,-1) $$

The foci are:

$$ (2-5,-1)=(-3,-1) $$

$$ (2+5,-1)=(7,-1) $$

The asymptotes are:

$$ y+1 = \pm \frac{4}{3}(x-2) $$

Example 2: Converting a General Equation to Standard Form

Consider this equation:

$$ 4x^2 - 9y^2 - 16x - 18y - 29 = 0 $$

Move the constant term and group like variables:

$$ 4x^2 - 16x - 9y^2 - 18y = 29 $$

Factor the squared-term coefficients from each group:

$$ 4(x^2-4x) - 9(y^2+2y) = 29 $$

Complete the square inside each group:

$$ 4\left((x-2)^2-4\right) - 9\left((y+1)^2-1\right) = 29 $$

Simplify:

$$ 4(x-2)^2 - 9(y+1)^2 = 36 $$

Divide by $36$:

$$ \frac{(x-2)^2}{9} - \frac{(y+1)^2}{4} = 1 $$

Now the structure is clear: the center is $(2,-1)$, the hyperbola opens left and right, $a=3$, and $b=2$.

Example 3: Using a Vertex and Focus from the Origin

Suppose an origin-centered hyperbola has one vertex at $(4,0)$ and one focus at $(5,0)$.

The distance from the origin to the vertex is:

$$ a=4 $$

The distance from the origin to the focus is:

$$ c=5 $$

Since $c^2=a^2+b^2$, solve for $b$:

$$ b=\sqrt{c^2-a^2}=\sqrt{5^2-4^2}=\sqrt{9}=3 $$

The standard equation is:

$$ \frac{x^2}{16} - \frac{y^2}{9} = 1 $$

This example works because the focus distance is greater than the vertex distance. If $c\le a$, the values do not form a real hyperbola of this type.

How to Interpret the Result

The standard equation is the main result because it organizes the hyperbola's geometry into a readable form.

The center $(h,k)$ tells you where the hyperbola is translated on the coordinate plane. If the center is $(0,0)$, the hyperbola is centered at the origin. If the center is not $(0,0)$, every vertex, focus, and asymptote is shifted around that center.

The values $a$, $b$, and $c$ have different meanings:

$a$ is the distance from the center to each vertex.
$b$ is the semi-conjugate axis value used for the guide rectangle and asymptote slopes.
$c$ is the distance from the center to each focus.

The transverse axis length is $2a$, not $a$. The conjugate axis length is $2b$, not $b$.

The eccentricity $e=\frac{c}{a}$ is always greater than $1$ for a valid hyperbola. A larger eccentricity means the focal distance $c$ is large compared with the vertex distance $a$.

The opening direction tells you where the two branches go. A horizontal hyperbola opens left and right. A vertical hyperbola opens up and down.

The vertices mark the closest points on the branches to the center along the transverse axis. The foci lie on the same transverse axis but farther from the center than the vertices.

The asymptotes show the long-run direction of the branches. The graph may show only a finite viewing window, but the mathematical hyperbola continues indefinitely.

Common Mistakes and Misconceptions

Confusing $a$ and $2a$. The value $a$ is a semi-axis length. The full transverse axis length is $2a$.

Thinking $a$ must be larger than $b$. For a hyperbola, $a$ is the semi-transverse axis and $b$ is the semi-conjugate axis. The meaning comes from the role each value plays, not from which number is larger.

Using the ellipse focal relation. Ellipses use a different relationship among $a$, $b$, and $c$. For hyperbolas, the relation is always:

$$ c^2=a^2+b^2 $$

Looking at the larger denominator to decide the opening direction. The opening direction comes from the positive squared term, not from the larger denominator.

Forgetting to complete the square correctly. In general form, any number added inside parentheses must be balanced after factoring the coefficient outside the parentheses.

Trying to use same-sign squared coefficients. In the supported general form, the $x^2$ and $y^2$ coefficients must have opposite signs. Same-sign squared coefficients usually point to an ellipse-type structure or another non-supported case, not the axis-aligned hyperbola form used here.

Trying to graph a rotated hyperbola as if it were axis-aligned. A rotated hyperbola usually has an $xy$ term. Standard horizontal and vertical forms do not handle that rotation directly.

Treating a sampled graph as the whole curve. A graphing window only shows part of the hyperbola. The branches continue beyond the displayed range and keep approaching the asymptotes.

When to Use Hyperbola Properties

Use hyperbola properties when you need to:

graph a hyperbola from its equation;
identify the center, vertices, foci, axes, and asymptotes;
convert a supported general equation into standard form;
check whether a conic equation is an axis-aligned hyperbola;
compare horizontal and vertical hyperbolas;
understand how $a$, $b$, $c$, and eccentricity shape the curve;
verify homework or study examples involving conic sections.

These properties are especially useful in algebra, precalculus, analytic geometry, and any course that studies conic sections.

Limitations and Things to Keep in Mind

The standard forms shown here describe axis-aligned hyperbolas. They do not directly describe rotated hyperbolas with an $xy$ term.

The general equation form used here assumes:

$$ Ax^2 + Cy^2 + Dx + Ey + F = 0 $$

with no $xy$ term. The coefficients $A$ and $C$ must be nonzero and have opposite signs for the supported hyperbola structure.

The values $a$ and $b$ must be positive. If either is $0$ or negative, the standard-form interpretation breaks down.

When using a vertex and focus from the origin, the focus distance must be greater than the vertex distance. In symbols:

$$ c>a>0 $$

This is required because:

$$ b=\sqrt{c^2-a^2} $$

A graph of a hyperbola is a visual aid, not a substitute for the equation. It may use rounded coordinates, sampled curve points, and a limited viewing window.

Decimal output should be treated as a formatted approximation when values are not exact. Very small values may be displayed as $0$, very large or very small nonzero values may use exponential notation, and trailing zeros may be removed. For exact symbolic work, keep radicals and fractions in your own written steps when needed.

If a hyperbola calculation affects graded work, engineering work, scientific modeling, or another important decision, double-check the setup and use a qualified source or instructor when appropriate.

How to Use This Calculator

Choose the input mode: standard, general, or foci.
For standard mode, enter the center values $h$ and $k$, the semi-transverse axis $a$, the semi-conjugate axis $b$, and the orientation.
For general mode, enter $A$, $C$, $D$, $E$, and $F$ for the supported equation $Ax^2 + Cy^2 + Dx + Ey + F = 0$.
For foci mode, enter one vertex and one focus on the same x-axis or y-axis relative to the origin. This mode treats the center as $(0,0)$ and does not infer rotated hyperbolas.
Review the standard equation, center, $a/b/c$ values, eccentricity, opening direction, vertices, foci, axis lengths, asymptotes, graph, and calculation steps.
Use an example button if you want to load a sample horizontal or vertical hyperbola.
Use the graph download option if you need to save the plotted hyperbola as a PNG.

Frequently Asked Questions

How do I know whether a hyperbola opens left/right or up/down?

Look for the positive squared term in standard form. If the $x$-term is positive, the hyperbola opens left and right. If the $y$-term is positive, it opens up and down.

What is the difference between $a$, $b$, and $c$?

The value $a$ is the distance from the center to each vertex. The value $b$ is the semi-conjugate axis value used in the asymptote slopes and guide rectangle. The value $c$ is the distance from the center to each focus.

Why is the hyperbola formula $c^2=a^2+b^2$?

For standard hyperbolas, the focal distance is related to the semi-transverse and semi-conjugate axis values by $c^2=a^2+b^2$. This makes $c$ greater than $a$, which places the foci outside the vertices along the transverse axis.

Can $b$ be larger than $a$?

Yes. In a hyperbola, $a$ means semi-transverse axis and $b$ means semi-conjugate axis. The labels describe their roles in the equation and graph, not which number is larger.

Can every hyperbola be entered in general mode?

No. The supported general mode is for axis-aligned equations of the form $Ax^2 + Cy^2 + Dx + Ey + F = 0$. Rotated hyperbolas with an $xy$ term are not included in that form.

Why does foci mode require the focus distance to be greater than the vertex distance?

For an origin-centered hyperbola built from a vertex distance $a$ and focus distance $c$, the semi-conjugate value is $b=\sqrt{c^2-a^2}$. If $c\le a$, then $b$ is not a positive real value, so the inputs do not define a valid hyperbola of this type.

Sources and References

Books and Open Textbooks

Jay Abramson. College Algebra 2e. OpenStax, 2021. Section 8.2, “The Hyperbola,” and the book preface/citation information. OpenStax Section 8.2
Carl Stitz and Jeff Zeager. Precalculus. 3rd Corrected Edition, Stitz Zeager Open Source Mathematics, 2013. Section 7.5, “Hyperbolas.” LibreTexts Section 7.5
Lynn Marecek and Andrea Honeycutt Mathis. Intermediate Algebra 2e. OpenStax, 2020. Section 11.4, “Hyperbolas.” OpenStax Section 11.4