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The Ellipse

Description And History of This Conic: An ellipse is composed of all points in a plane whose distances from two fixed points, called the foci, when combined, equals a constant. In geometric terms, an ellipse is created when a slightly tilted plane intersects each generator of only one nappe of a cone. For a clear picture of an ellipse, refer to the diagram on the homepage.

The Basics

Here is a diagram of basic ellipse terms to help you better understand its description. First, the main definition expresses an ellipse as having the distance from a point to two fixed points, the foci, equaling a constant. This is displayed, though not to scale, as the pink lines, emanating from both foci (green dots), to the point (brown dot). Their combined distance equals what we will address next, the length of the major axis (the thick red line). As you see, one length of the ellipse is bigger than the other (in this case, its length along the x-axis is longer than that of its height along the y-axis). This longer side lies along what we call the major axis, a line that contains the foci. The minor axis is a line through the center and perpendicular to the major axis. The two points where the major axis intersects the ellipse, noted by red dots, are the vertices (the places where the minor axis intersects the ellipse are also called vertices, though we do not usually address them). The distance from one vertex to the other is called the length of the major axis.

The Equations

As we now know, the basic equation for all conics is "Ax^2 + Cy^2 + Dx + Ey + F = 0." Since an ellipse has two squared variables, x and y, you can identify an ellipse easily by knowing that "AC > 0." Here's why: The normal equation for an ellipse is "(x-h)^2/a^2) + (y-k)^2/b^2) = 1" and since x and y are being added to equal 1, A and C will be positive values, thus when multiplied, will be greater than zero. The variable "a" in the equation is the distance from the center to either major axis vertex, while the variable "b" is the distance from the center to either minor axis vertex. A new variable, "c", is used in this conic and represents the distance from the center to either foci. To solve for any of the above variables when working on a problem set, keep in mind this equation: b^2 = a^2 - c^2. In the diagram to the right, we happen to have our major axis along the x-axis, and thus our "a" variables as well. Another situation can arise if our major axis lies along the y value, making it appear taller than wide. In this case, the equation stays the same, but the "a" and "b" variables switch places like this: "(x-h)^2/b^2) + (y-k)^2/a^2) = 1."

Special Cases: Ellipse

Since the ellipse is a conic just like the parabola, certain special cases can arise under the right circumstances. As before, the first case is the rotation of axes. For review, when there is rotation of axes, the origin of the conic remains the same, while the x and y axes are rotated through a given angle to a new position, with new variables x' and y'. In this case, the normal equation for a conic changes from "Ax^2 + Cy^2 + Dx + Ey + F = 0" to "Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0", thus adding "Bxy". To identify an ellipse, we once again employ the discriminant, "B^2 - 4AC", and if our answer is less than zero, we have an ellipse because the ellipse is defined as "B^2 - 4AC<0." *************************************************************************************** Another property we will once again explore is that of eccentricity. Eccentricity is defined as the ratio of the distance (d) from the focus (F) to a given point (P) in a conic to that of the distance (d) from the directrix (D) to the point (P). Look to the diagram on the right to see this definition written as an equation. Eccentricity is also defined in simpler terms as "c/a." In the case of this conic, an ellipse is defined as "eccentricity < 1" (e < 1). This is obvious if we keep in mind the fact that since the distance from the center to the outer vertex is the variable "a", while the center to the foci, which is a shorter distance than to the vertex, is the variable "c", your ratio will be less than 1.