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Hyperbolas

Description And History of This Conic: A Hyperbola is the collection of all points in a plane the difference of whose distances from two fixed points, called the foci, equals a positive constant. In Geometric terms, a hyperbola is made when a plane intersects both nappes of a double-napped cone. Return to the homepage to view an accurate diagram of a hyperbola.

Basics

Here is a diagram of basic hyperbola terms so that you can better understand the seemingly confusing properties of this conic. First, let's tackle the wordy definition of a hyperbola. Taking a look at the two pink line segments, starting from either foci (the pink dots) and ending at a representative point "P", the distance of their difference (subtract them) equals a constant value. The constant is equal to the distance between the two vertices (green dots), just like the constant value of the ellipse. If you notice the axes of the diagram, there are two new terms: Transverse axis and conjugate axis. The transverse axis is a line containing the foci of the hyperbola, while the conjugate axis is a line through the center of the conic and perpendicular to the transverse axis. In this diagram, the transverse axis appears on the x-axis, and the conjugate on the y-axis, though you will soon learn that they can reverse positioning depending on the position of the hyperbola.

The Equations

By this time, we definitely know that the equation for all conics is "Ax^2 + Cy^2 + Dx + Ey + F = 0." Since a hyperbola, like an ellipse, has two squared variables, x and y, you can identify a hyperbola easily by knowing that "AC < 0." Here's why: the normal equation for a hyperbola is "(x-h)^2/a^2) - (y-k)^2/b^2) = 1" for a hyperbola parallel to the x-axis, and it becomes "(y-k)^2/a^2) - (x-h)^2/b^2) = 1" for a hyperbola parallel to the y-axis. Since there is subtraction of the squared x and y variables, either A or C will result in a negative number, as in this example of a hyperbola: 2x^2- 3y^2 + 6y +4= 0. In this case, A= 2 while C= -3, thus AC= -6 which is less than zero. Something to note about the hyperbola equation is the fact that the squared variable "a", which is the distance from the center to a vertex (along the transverse axis), does not move, but instead notifies us to which axis our hyperbola will be parallel. The other variables, b and c, represent the distance to a conjugate axis vertex and the distance from the center to a focus, respectively. The last equation we should discuss is that of the asymptotes for a hyperbola. If parallel to the x-axis, the equation follows as: y - k = + b/a (x - h). If parallel to the y-axis, the equation changes slightly: y - k = + a/b (x - h). These are represented by the dashed blue lines that intersect at the center of the hyperbola.

Special Cases: Hyperbola

Since the hyperbola is like most conics, 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 a hyperbola, we once again employ the discriminant, "B^2 - 4AC", and if our answer is greater than zero, we have an hyperbola because the hyperbola 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). 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 vertex is the variable "a", while the center to the foci, which is a longer distance than to the vertex, is the variable "c", your ratio will be greater than 1.