Hyperbola Equation Calculator – Standard, General & Asymptote Forms
A hyperbola is the set of points \(P\) in the plane such that the
absolute difference of distances to two fixed points (the foci) is constant:
\[
\big|PF_1 - PF_2\big| = 2a.
\]
This tool focuses on axis-aligned hyperbolas (no rotation / no \(xy\) term).
1) Standard (center) forms
The standard forms show the center, transverse axis direction, and parameters immediately.
-
Horizontal transverse axis (opens left and right):
\[
\frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2}=1
\]
Center \(C=(h,k)\). Vertices \(V=(h\pm a,k)\). Conjugate endpoints \(B=(h,k\pm b)\).
-
Vertical transverse axis (opens up and down):
\[
\frac{(y-k)^2}{a^2}-\frac{(x-h)^2}{b^2}=1
\]
Center \(C=(h,k)\). Vertices \(V=(h,k\pm a)\). Conjugate endpoints \(B=(h\pm b,k)\).
Here \(a>0\) is the transverse radius (distance from center to a vertex),
and \(b>0\) is the conjugate radius.
2) Foci, eccentricity, and key distances
For both orientations:
\[
c=\sqrt{a^2+b^2},\qquad e=\frac{c}{a}\quad (e>1).
\]
-
Horizontal: foci \(F=(h\pm c,k)\)
-
Vertical: foci \(F=(h,k\pm c)\)
3) Asymptotes and the guiding rectangle
A hyperbola approaches (but never touches) its two asymptote lines. The simplest way to remember them is:
“remove the \(1\)” from the standard equation.
-
Horizontal:
\[
\frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2}=1
\quad\Longrightarrow\quad
y-k=\pm\frac{b}{a}(x-h)
\]
-
Vertical:
\[
\frac{(y-k)^2}{a^2}-\frac{(x-h)^2}{b^2}=1
\quad\Longrightarrow\quad
y-k=\pm\frac{a}{b}(x-h)
\]
The guiding rectangle is centered at \((h,k)\). Its corners are:
- Horizontal: \((h\pm a,k\pm b)\)
- Vertical: \((h\pm b,k\pm a)\)
The asymptotes pass through the rectangle’s corners. This rectangle is a helpful visual aid when sketching.
4) General form → standard form (complete the square)
An axis-aligned hyperbola can be written as:
\[
A x^2 + C y^2 + D x + E y + F = 0,
\]
where \(A\) and \(C\) must have opposite signs:
\[
A\cdot C < 0.
\]
Step 1: compute the center by matching the square-completion pattern:
\[
h=-\frac{D}{2A},\qquad k=-\frac{E}{2C}.
\]
Step 2: completing the squares gives:
\[
A(x-h)^2 + C(y-k)^2 = R,
\quad\text{where}\quad
R=Ah^2+Ck^2-F.
\]
Step 3: divide by \(R\) and match standard form.
If \(R<0\), multiplying the entire equation by \(-1\) flips the sign of \(R\) without changing the curve.
After normalization with \(R>0\), one of the squared terms will have a positive coefficient and the other negative.
The positive term determines the transverse axis direction.
5) Extra value mode: center + asymptote slope + point
Asymptotes alone determine only the ratio \(b/a\) (or \(a/b\)). If you also know one point
\(P(x_0,y_0)\) on the hyperbola, you can solve for \(a\) and \(b\).
6) Worked example
Example:
\[
\frac{x^2}{9}-\frac{y^2}{4}=1
\]
This is a horizontal hyperbola with center \((0,0)\), \(a^2=9\Rightarrow a=3\), \(b^2=4\Rightarrow b=2\).
Foci:
\[
c=\sqrt{a^2+b^2}=\sqrt{9+4}=\sqrt{13}
\quad\Longrightarrow\quad
F=(\pm\sqrt{13},0).
\]
Asymptotes:
\[
y=\pm\frac{b}{a}x=\pm\frac{2}{3}x.
\]
Graph note: The calculator plot uses square units (equal scaling) so the hyperbola shape
is not distorted, and tick labels are drawn near their axes for readability while panning/zooming.
