An ellipse is the set of all points in a plane such that the sum of the distances from two fixed points, called the foci (singular: focus), is constant. This constant sum is equal to the length of the major axis (2a). This geometric property creates a shape resembling a stretched or flattened circle.
| Term | Symbol | Description |
|---|---|---|
| Center | (h, k) | The midpoint of the major and minor axes. |
| Semi-major Axis | a | Half the length of the longest diameter of the ellipse. |
| Semi-minor Axis | b | Half the length of the shortest diameter of the ellipse. |
| Foci | F₁, F₂ | The two fixed points used to define the ellipse. |
| Focal Distance | c | The distance from the center to each focus. |
| Vertices | The endpoints of the major axis. | |
| Co-vertices | The endpoints of the minor axis. | |
| Eccentricity | e | A measure of how elongated the ellipse is, ranging from 0 (a perfect circle) to just under 1. |
An ellipse is centered at point (h, k). The major axis is the longest diameter, with length 2a, and its endpoints are the vertices. The minor axis is the shortest diameter, with length 2b, and its endpoints are the co-vertices. The two foci, F₁ and F₂, lie on the major axis, each at a distance 'c' from the center. The semi-major axis 'a' is the distance from the center to a vertex, and the semi-minor axis 'b' is the distance from the center to a co-vertex.
Symmetry: An ellipse is symmetric with respect to its major axis, its minor axis, and its center.
Focal Definition: The sum of the distances from any point on the ellipse to the two foci is constant and equal to the length of the major axis (2a). This is the fundamental defining property.
Reflection Property: A ray of light or sound originating at one focus will reflect off the ellipse and pass through the other focus. This is the principle behind whispering galleries.
Eccentricity: The eccentricity 'e' (where 0 ≤ e < 1) determines the shape of the ellipse. If e = 0, the ellipse is a circle. As 'e' approaches 1, the ellipse becomes increasingly elongated.
Boundaries: All points of the ellipse are contained within a rectangle of dimensions 2a by 2b.
The standard equation of an ellipse can be derived from its definition using the distance formula. We start with the focal property: the sum of the distances from any point P(x, y) on the ellipse to the two foci is a constant, 2a.
Step 1: Set up the equation. Let the foci be placed on the x-axis at F₁(-c, 0) and F₂(c, 0). Let P(x, y) be any point on the ellipse. The defining property is:
Using the distance formula, this becomes:
Step 2: Isolate a radical and square both sides.
Step 3: Simplify and isolate the remaining radical. Expand the squared terms and cancel common terms.
Step 4: Square both sides again and simplify.
Step 5: Substitute b² for a² - c². From the relationship between the axes and focal distance, we know that c² = a² - b², or b² = a² - c².
Step 6: Divide by a²b² to get the standard form.
The orbits of planets, moons, comets, and satellites are ellipses, as described by Kepler's laws of planetary motion. Engineers use elliptical equations to calculate spacecraft trajectories for interplanetary missions.
The reflective property of ellipses is used in 'whispering galleries'. In an elliptical room, a sound made at one focus will be clearly heard at the other focus. This principle is used in designing auditoriums and concert halls.
A medical procedure called lithotripsy uses an elliptical reflector to treat kidney stones. High-energy shock waves are generated at one focus and concentrated at the other focus, where the kidney stone is placed, breaking it apart without surgery.
Elliptical reflectors are used to focus light and other electromagnetic waves. They are found in car headlights, telescope mirrors, and antennas to direct a beam with high efficiency.
Planetary Orbits
Every planet, asteroid, and comet in our solar system follows an elliptical path around the sun. This predictable motion allows astronomers to forecast events like eclipses and meteor showers and to plan space missions.
Whispering Galleries
In architectural spaces with elliptical domes, such as St. Paul's Cathedral in London or Statuary Hall in the U.S. Capitol, a person whispering at one focus can be heard clearly by someone at the other focus due to the reflection property of the ellipse.
Product Design
The shape of an ellipse is often used in design for both aesthetic and functional reasons. This includes elliptical dining tables, running tracks, mirrors, and logos, which provide a visually appealing and often ergonomic form.
Ellipses are classified based on their eccentricity (e), which describes their shape.
| Type | Eccentricity (e) | Description |
|---|---|---|
| Circle | e = 0 | A special case where both foci are at the center. The semi-major and semi-minor axes are equal (a = b). |
| Standard Ellipse | 0 < e < 1 | The typical oval shape where the foci are distinct. As 'e' approaches 0, the ellipse becomes more circular. As 'e' approaches 1, it becomes more elongated. |
| Degenerate Ellipse (Line Segment) | e = 1 | A theoretical limit where the ellipse flattens completely into a line segment of length 2a, connecting the two foci. |
Confusing 'a' and 'b': Always remember that 'a' represents the semi-major axis and is always the larger value. The orientation of the ellipse (horizontal or vertical) depends on whether a² is under the x² or y² term, not on which variable comes first.
Incorrect Focal Distance Formula: A frequent error is using c² = a² + b² (for hyperbolas) instead of the correct formula for ellipses: c² = a² - b². An easy way to remember is that the foci must be inside the vertices, so c must be smaller than a.
Misplacing the Foci: After calculating 'c', be sure to place the foci along the major axis. If the ellipse is horizontal (wider than it is tall), the foci are at (h ± c, k). If it is vertical (taller than it is wide), the foci are at (h, k ± c).