Conic Section Foci Calculator – For Ellipse, Hyperbola, Parabola
A focus (plural: foci) is a special point that helps define a conic section.
This tool works with axis-aligned conics (no rotation, i.e. no \(xy\) term).
It finds the foci from standard parameters and also visualizes them on the graph.
1) Ellipse foci
An ellipse can be written (axis-aligned) as:
\[
\frac{(x-h)^2}{D_x}+\frac{(y-k)^2}{D_y}=1,
\qquad D_x>0,\;D_y>0.
\]
The semi-axis lengths are:
\[
a_x=\sqrt{D_x},\qquad a_y=\sqrt{D_y}.
\]
Let \(a=\max(a_x,a_y)\) and \(b=\min(a_x,a_y)\). Then the focus distance from the center is:
\[
c=\sqrt{a^2-b^2}.
\]
-
If \(a=a_x\) (major axis is horizontal), the foci are:
\[
(h\pm c,\;k).
\]
-
If \(a=a_y\) (major axis is vertical), the foci are:
\[
(h,\;k\pm c).
\]
If \(a=b\), the ellipse is a circle and \(c=0\) (the two foci coincide at the center).
2) Hyperbola foci
A hyperbola has two common axis-aligned standard forms:
-
Horizontal transverse axis
\[
\frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2}=1.
\]
Foci:
\[
(h\pm c,\;k),
\qquad c=\sqrt{a^2+b^2}.
\]
Asymptotes:
\[
y-k=\pm\frac{b}{a}(x-h).
\]
-
Vertical transverse axis
\[
\frac{(y-k)^2}{a^2}-\frac{(x-h)^2}{b^2}=1.
\]
Foci:
\[
(h,\;k\pm c),
\qquad c=\sqrt{a^2+b^2}.
\]
Asymptotes:
\[
y-k=\pm\frac{a}{b}(x-h).
\]
3) Parabola focus and directrix
A parabola is the set of all points whose distance to a fixed point (the focus)
equals its distance to a fixed line (the directrix).
Axis-aligned standard forms are:
-
Vertical axis
\[
(x-h)^2=4p(y-k).
\]
Vertex \(V=(h,k)\), focus \(F=(h,k+p)\), directrix \(y=k-p\), axis \(x=h\).
-
Horizontal axis
\[
(y-k)^2=4p(x-h).
\]
Vertex \(V=(h,k)\), focus \(F=(h+p,k)\), directrix \(x=h-p\), axis \(y=k\).
The parameter \(p\) is signed: \(p>0\) opens upward/right and \(p<0\) opens downward/left.
4) Eccentricity (optional extra)
Eccentricity measures how “stretched” a conic is:
- Ellipse: \(\;0\le e<1,\quad e=\dfrac{c}{a}\).
- Parabola: \(\;e=1\).
- Hyperbola: \(\;e>1,\quad e=\dfrac{c}{a}\).
Example (ellipse)
Given:
\[
\frac{x^2}{16}+\frac{y^2}{9}=1
\]
Here \(h=0\), \(k=0\), \(D_x=16\Rightarrow a_x=4\), \(D_y=9\Rightarrow a_y=3\).
So \(a=4\), \(b=3\), and:
\[
c=\sqrt{a^2-b^2}=\sqrt{16-9}=\sqrt{7}.
\]
Major axis is horizontal, so the foci are:
\[
(\pm\sqrt{7},\;0).
\]
On the calculator graph, the foci are marked and labeled (e.g., \(F1\), \(F2\), or \(F\)).
