Theory: Hyperbola Eccentricity Calculator
A hyperbola is the set of points \(P\) in the plane whose absolute difference of distances
to two fixed points (the foci) is constant. It has two branches and
two asymptotes. In this tool we focus on axis-aligned hyperbolas
(no rotation, so there is no \(xy\) term).
1) Standard forms (center \((h,k)\))
Hyperbolas come in two orientations depending on which variable has the positive squared term.
-
Horizontal hyperbola (opens left/right):
\[
\frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2}=1
\]
The transverse axis is horizontal and \(a>0,\,b>0\).
-
Vertical hyperbola (opens up/down):
\[
\frac{(y-k)^2}{a^2}-\frac{(x-h)^2}{b^2}=1
\]
The transverse axis is vertical and \(a>0,\,b>0\).
The number \(a\) controls how far the vertices are from the center; \(b\) controls how “wide” the branches open.
2) Eccentricity for a hyperbola
Define
\[
c=\sqrt{a^2+b^2}.
\]
Then the eccentricity is
\[
e=\frac{c}{a}=\sqrt{1+\frac{b^2}{a^2}}.
\]
For every hyperbola, \(e>1\). Larger \(e\) generally means the branches are “closer” to the asymptotes.
3) Key geometric features
Vertices
-
Horizontal:
\[
V_1=(h-a,k),\quad V_2=(h+a,k)
\]
-
Vertical:
\[
V_1=(h,k-a),\quad V_2=(h,k+a)
\]
Foci
-
Horizontal:
\[
F_1=(h-c,k),\quad F_2=(h+c,k)
\]
-
Vertical:
\[
F_1=(h,k-c),\quad F_2=(h,k+c)
\]
Asymptotes
The asymptotes describe the directions the branches approach as \(|x|\) or \(|y|\) grows.
-
Horizontal:
\[
y-k=\pm\frac{b}{a}(x-h)
\]
-
Vertical:
\[
y-k=\pm\frac{a}{b}(x-h)
\]
Directrices
The directrices are the lines used in the focus-directrix definition of a conic.
For a hyperbola, they are:
-
Horizontal:
\[
x=h\pm\frac{a}{e}
\]
-
Vertical:
\[
y=k\pm\frac{a}{e}
\]
4) Converting a general equation (no rotation)
A non-rotated hyperbola can be written as:
\[
Ax^2 + Cy^2 + Dx + Ey + F = 0
\]
with opposite signs for the quadratic coefficients: \(A\cdot C<0\).
Completing the square gives:
\[
A(x-h)^2 + C(y-k)^2 = K,
\quad h=-\frac{D}{2A},\quad k=-\frac{E}{2C}.
\]
After dividing by \(K\), you obtain a standard form where the positive term identifies the orientation.
The calculator automates these steps and then computes \(a,b,c,e\), foci, asymptotes, and directrices.
5) Worked example
Given \(a=2\) and \(b=3\):
-
Compute
\[
c=\sqrt{a^2+b^2}=\sqrt{4+9}=\sqrt{13}.
\]
-
Eccentricity:
\[
e=\frac{c}{a}=\frac{\sqrt{13}}{2}=\sqrt{\frac{13}{4}}\approx 1.8028.
\]
6) Tips and common mistakes
-
Make sure \(a>0\) and \(b>0\). If you enter negative values, use absolute values or correct your input.
-
In standard form, the denominator under the positive squared term is \(a^2\) (transverse direction).
-
For general equations, if \(A\cdot C>0\), the conic is not a hyperbola (it could be an ellipse/circle).
-
Remember: hyperbola eccentricity satisfies \(e>1\). If you get \(e\le 1\), the inputs do not describe a real hyperbola.
