Table of Contents
Geometric Definition and Basic Features
An ellipse is the set of all points in a plane whose distances to two fixed points (called foci, singular: focus) have a constant sum.
If the foci are $F_1$ and $F_2$, and $P$ is any point on the ellipse, then
$$
PF_1 + PF_2 = \text{constant}.
$$
From this definition:
- The two foci lie along a central line of symmetry.
- The shortest distance across the ellipse (through its center) is the minor axis.
- The longest distance across (through its center) is the major axis.
- The midpoint of the segment connecting the foci is the center of the ellipse.
On each axis:
- The endpoints of the major axis are called vertices.
- The endpoints of the minor axis are sometimes called co-vertices.
Standard Equations of Ellipses
Throughout, assume $a>0$ and $b>0$ are real numbers related to the sizes of the axes.
Horizontal and Vertical Ellipses (Centered at the Origin)
- Horizontal major axis
$$
\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1,\quad a>b>0.
$$
- Center: $(0,0)$.
- Major axis is horizontal.
- Vertices: $(\pm a, 0)$.
- Co-vertices: $(0, \pm b)$.
- The length of the major axis is $2a$.
- The length of the minor axis is $2b$.
- Vertical major axis
$$
\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1,\quad a>b>0.
$$
- Center: $(0,0)$.
- Major axis is vertical.
- Vertices: $(0, \pm a)$.
- Co-vertices: $(\pm b, 0)$.
- Major axis length: $2a$.
- Minor axis length: $2b$.
In both cases, $a$ is half the length of the major axis, and $b$ is half the length of the minor axis.
Ellipses Centered at $(h,k)$
Shifting the center from the origin to $(h,k)$ gives:
- Horizontal major axis
$$
\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1,\quad a>b>0.
$$
- Center: $(h,k)$.
- Vertices: $(h \pm a,\, k)$.
- Co-vertices: $(h,\, k \pm b)$.
- Vertical major axis
$$
\frac{(x-h)^2}{b^2} + \frac{(y-k)^2}{a^2} = 1,\quad a>b>0.
$$
- Center: $(h,k)$.
- Vertices: $(h,\, k \pm a)$.
- Co-vertices: $(h \pm b,\, k)$.
Relationship Between $a$, $b$, and $c$
The distance from the center to each focus is usually denoted by $c$.
For an ellipse with major axis along the $x$-axis (horizontal) or $y$-axis (vertical), the foci lie on the major axis, symmetrically placed about the center. The three quantities $a$, $b$, and $c$ satisfy
$$
c^2 = a^2 - b^2,\quad 0 < c < a.
$$
Thus
$$
c = \sqrt{a^2 - b^2}.
$$
Foci for Standard Ellipses
- Horizontal ellipse, centered at $(0,0)$
$$
\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1,\quad a>b>0.
$$ - Foci: $(\pm c, 0)$ where $c = \sqrt{a^2 - b^2}$.
- Vertical ellipse, centered at $(0,0)$
$$
\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1,\quad a>b>0.
$$ - Foci: $(0, \pm c)$ where $c = \sqrt{a^2 - b^2}$.
- Horizontal ellipse, centered at $(h,k)$
$$
\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1.
$$ - Foci: $(h \pm c,\, k)$, where $c = \sqrt{a^2 - b^2}$.
- Vertical ellipse, centered at $(h,k)$
$$
\frac{(x-h)^2}{b^2} + \frac{(y-k)^2}{a^2} = 1.
$$ - Foci: $(h,\, k \pm c)$, where $c = \sqrt{a^2 - b^2}$.
Eccentricity of an Ellipse
The eccentricity of an ellipse, denoted $e$, measures how “stretched” it is:
$$
e = \frac{c}{a},\quad 0 < e < 1.
$$
Using $c^2 = a^2 - b^2$, we can write
$$
e = \frac{\sqrt{a^2 - b^2}}{a}.
$$
Some observations:
- If $e$ is close to $0$, the ellipse is nearly circular ($b$ close to $a$).
- If $e$ is close to $1$, the ellipse is very elongated ($b$ much smaller than $a$).
- A circle can be viewed as a special case of an ellipse with $a=b$, which gives $c=0$ and $e=0$.
Graphing Ellipses from Their Equations
To sketch an ellipse from a standard-form equation:
- Identify the center $(h,k)$ by looking at the terms $(x-h)^2$ and $(y-k)^2$.
- Identify $a^2$ and $b^2$ from the denominators.
- The larger denominator corresponds to $a^2$ (major axis).
- The smaller denominator corresponds to $b^2$ (minor axis).
- Determine orientation:
- If $a^2$ is under $(x-h)^2$, the major axis is horizontal.
- If $a^2$ is under $(y-k)^2$, the major axis is vertical.
- Plot vertices and co-vertices:
- Move $a$ units from the center along the major axis to get vertices.
- Move $b$ units from the center along the minor axis to get co-vertices.
- Sketch the ellipse as a smooth, symmetric curve through these four points.
If you also want the foci, compute
$$
c = \sqrt{a^2 - b^2}
$$
and place them on the major axis at distance $c$ from the center.
Recognizing an Ellipse from a General Quadratic Equation
A general second-degree equation in $x$ and $y$ has the form
$$
Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0.
$$
In this chapter we restrict attention to non-rotated ellipses, where $B=0$ and the axes of the ellipse are aligned with the coordinate axes. In that case, the equation simplifies to
$$
Ax^2 + Cy^2 + Dx + Ey + F = 0.
$$
For such an equation to represent an ellipse (not degenerate):
- $A$ and $C$ have the same sign (both positive or both negative),
- and $A \neq C$ (or else it is a circle, in standard position).
To put it into standard form, you complete the square separately in $x$ and $y$, then divide through so the right-hand side becomes $1$. That process yields one of the standard forms already given, revealing the center, $a$, $b$, orientation, and so on.
Area of an Ellipse
For an ellipse with semi-axes $a$ and $b$ (major semi-axis $a$, minor semi-axis $b$), its area is
$$
\text{Area} = \pi a b.
$$
This holds regardless of where the ellipse is centered; only the lengths of the semi-axes matter.
Brief Note on Rotated Ellipses
If an ellipse is rotated, its equation typically includes an $xy$-term:
$$
Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0,\quad B\neq 0.
$$
Such ellipses are not aligned with the coordinate axes. Analyzing them generally involves a rotation of the coordinate system to eliminate the $xy$-term; that process is treated elsewhere in analytic geometry. Here we focus on ellipses whose axes are parallel to the coordinate axes.