Ellipse Equation Calculator – Standard, General & Vertex Forms
An ellipse is the set of points whose distances to two fixed points (the foci) add to a constant.
In coordinate geometry, an axis-aligned ellipse is written in standard (center) form:
\[
\frac{(x-h)^2}{a_x^2}+\frac{(y-k)^2}{b_y^2}=1,
\]
where \((h,k)\) is the center and \(a_x,b_y>0\) are the semi-axis lengths along the \(x\)- and \(y\)-directions.
1) Major axis, minor axis, and foci
Let
\[
a=\max(a_x,b_y), \qquad b=\min(a_x,b_y).
\]
The major axis is the direction of \(a\), and the minor axis is the direction of \(b\).
The focal distance is
\[
c=\sqrt{a^2-b^2},
\]
and the eccentricity is
\[
e=\frac{c}{a}\quad (0\le e<1).
\]
If the major axis is horizontal, the foci are \((h\pm c,k)\).
If it is vertical, the foci are \((h,k\pm c)\).
2) General form (axis-aligned)
An axis-aligned ellipse can be written (without an \(xy\) term) as:
\[
A x^2 + C y^2 + D x + E y + F = 0,
\]
with \(A>0\) and \(C>0\). To convert to standard form, we complete the square.
The center is:
\[
h=-\frac{D}{2A},\qquad k=-\frac{E}{2C}.
\]
After completing squares:
\[
A(x-h)^2 + C(y-k)^2 = R,\quad\text{where}\quad R=Ah^2+Ck^2-F.
\]
For a real ellipse, \(R>0\). Dividing by \(R\) gives:
\[
\frac{(x-h)^2}{R/A}+\frac{(y-k)^2}{R/C}=1,
\qquad a_x^2=\frac{R}{A},\quad b_y^2=\frac{R}{C}.
\]
3) Standard/vertex parameters → general form
Starting from
\[
\frac{(x-h)^2}{a_x^2}+\frac{(y-k)^2}{b_y^2}=1,
\]
multiply by \(a_x^2 b_y^2\) and expand to obtain:
\[
A x^2 + C y^2 + D x + E y + F = 0,
\]
where the coefficients depend on \(h,k,a_x,b_y\). The calculator performs this expansion automatically.
4) Vertices + co-vertices → ellipse equation
If you know the four key points of an axis-aligned ellipse:
- Vertices \(V_1,V_2\): endpoints of the major axis
- Co-vertices \(CV_1,CV_2\): endpoints of the minor axis
Then the center is the midpoint of either pair:
\[
(h,k)=\left(\frac{x_{V1}+x_{V2}}{2},\frac{y_{V1}+y_{V2}}{2}\right)
=\left(\frac{x_{CV1}+x_{CV2}}{2},\frac{y_{CV1}+y_{CV2}}{2}\right).
\]
For axis-aligned ellipses, the vertices share either the same \(y\) (horizontal major axis) or the same \(x\) (vertical major axis).
The radii are:
\[
a=\frac{\lVert V_1V_2\rVert}{2},\qquad b=\frac{\lVert CV_1CV_2\rVert}{2}.
\]
Finally, assign \(a_x\) and \(b_y\) depending on orientation:
• Horizontal major axis: \(a_x=a\), \(b_y=b\)
• Vertical major axis: \(a_x=b\), \(b_y=a\)
Graph note
The plot uses square units (equal scaling on both axes), so the ellipse is not visually distorted.
Tick labels are drawn near their axes for readability while panning/zooming.
