Hyperbola Properties Calculator – Foci, Asymptotes & Eccentricity
A hyperbola is the set of points \(P\) in the plane for which the
absolute difference of distances to two fixed points (the foci) is constant.
This tool focuses on axis-aligned hyperbolas (no rotation, so no \(xy\) term).
1) Standard forms and orientation
With center \((h,k)\), semi-transverse axis \(a>0\), and semi-conjugate axis \(b>0\):
-
Horizontal transverse axis (opens left/right):
\[
\frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2}=1
\]
Vertices: \((h\pm a,k)\).
-
Vertical transverse axis (opens up/down):
\[
\frac{(y-k)^2}{a^2}-\frac{(x-h)^2}{b^2}=1
\]
Vertices: \((h,k\pm a)\).
2) Foci and eccentricity
Define
\[
c=\sqrt{a^2+b^2}.
\]
Then the foci are:
-
Horizontal: \(F_{1,2}=(h\pm c,k)\)
-
Vertical: \(F_{1,2}=(h,k\pm c)\)
The eccentricity is
\[
e=\frac{c}{a} \quad (>1\ \text{for all hyperbolas}).
\]
3) Asymptotes (most important property for the graph)
The asymptotes are the lines that the hyperbola approaches at large \(|x|\) or \(|y|\).
Starting from the standard form and dropping the “\(=1\)” term yields:
-
Horizontal:
\[
y-k=\pm\frac{b}{a}(x-h)
\]
-
Vertical:
\[
y-k=\pm\frac{a}{b}(x-h)
\]
A useful construction is the guiding rectangle centered at \((h,k)\):
• Horizontal: corners \((h\pm a,\ k\pm b)\)
• Vertical: corners \((h\pm b,\ k\pm a)\)
The asymptotes pass through the rectangle’s diagonals.
4) Directrices and latus rectum
The directrices for a hyperbola are:
-
Horizontal:
\[
x=h\pm \frac{a}{e}=h\pm\frac{a^2}{c}
\]
-
Vertical:
\[
y=k\pm \frac{a}{e}=k\pm\frac{a^2}{c}
\]
The latus rectum is the chord through a focus perpendicular to the transverse axis.
Its length is
\[
\frac{2b^2}{a}.
\]
5) Converting a general equation (no rotation)
A non-rotated conic can be written as:
\[
Ax^2+Cy^2+Dx+Ey+F=0
\]
(no \(Bxy\) term). Completing the square gives a centered form:
\[
A(x-h)^2 + C(y-k)^2 = \text{constant}
\]
where
\[
h=-\frac{D}{2A},\qquad k=-\frac{E}{2C}.
\]
After dividing by the constant, a hyperbola is identified by the presence of
opposite signs on the squared terms.
Example
For
\[
\left(\frac{x}{5}\right)^2-\left(\frac{y}{4}\right)^2=1
\quad\Longleftrightarrow\quad
\frac{x^2}{25}-\frac{y^2}{16}=1,
\]
we have \(h=k=0\), \(a=5\), \(b=4\), so:
- \(c=\sqrt{a^2+b^2}=\sqrt{25+16}=\sqrt{41}\)
- Foci: \((\pm\sqrt{41},0)\)
- Asymptotes: \(y=\pm\frac{4}{5}x\)
- Eccentricity: \(e=\dfrac{\sqrt{41}}{5}\)
