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Example:Equation of Hyperbola

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Description: The image shows a double cone in which a geometrical plane has sliced off parts of the top and bottom half; the boundary curve of the slice on the cone is the hyperbola. A double cone consists of two cones stacked point-to-point and sharing the same axis of rotation; it may be generated by rotating a line about an axis that passes through a point of the line.


A hyperbola is an open curve with two branches, the intersection of a plane with both halves of a double cone.


In mathematics a hyperbola is a curve, specifically a smooth curve that lies in a plane, which can be defined either by its geometric properties or by the kinds of equations for which it is the solution set. A hyperbola has two pieces, called connected components or branches, which are mirror images of each other and resembling two infinite bows. The hyperbola is one of the four kinds of conic section, formed by the intersection of a plane and a cone. The other conic sections are the parabola, the ellipse, and the circle (the circle is a special case of the ellipse). Which conic section is formed depends on the angle the plane makes with the axis of the cone, compared with the angle a line on the surface of the cone makes with the axis of the cone. If the angle between the plane and the axis is less than the angle between the line on the cone and the axis, or if the plane is parallel to the axis, then the conic is a hyperbola.


Nomenclature and features

The asymptotes of the hyperbola (red curves) are shown as blue dashed lines and intersect at the center of the hyperbola, C. The two focal points are labeled F1 and F2, and the thin black line joining them is the transverse axis. The perpendicular thin black line through the center is the conjugate axis. The two thick black lines parallel to the conjugate axis (thus, perpendicular to the transverse axis) are the two directrices, D1 and D2. The eccentricity e equals the ratio of the distances from a point P on the hyperbola to one focus and its corresponding directrix line (shown in green). The two vertices are located on the transverse axis at ±a relative to the center. So the parameters are: a - distance from center C to either vertex
b - length of a perpendicular segment from each vertex to the asymptotes
c - distance from center C to either Focus point, F1 and F2, and
θ - angle formed by each asymptote with the transverse axis.


 A hyperbola is an open curve, meaning that it continues indefinitely to infinity, rather than closing on itself as an ellipse does. A hyperbola consists of two disconnected curves called its arms or branches.

The points on the two branches that are closest to each other are called their vertices, and the line segment connecting them is called the transverse axis or major axis, corresponding to the major diameter of an ellipse. The midpoint of the transverse axis is known as the hyperbola's center. The distance a from the center to each vertex is called the semi-major axis. Outside of the transverse axis but on the same line are the two focal points (foci) of the hyperbola. The line through these five points is one of the two principal axes of the hyperbola, the other being the perpendicular bisector of the transverse axis. The hyperbola has mirror symmetry about its principal axes, and is also symmetric under a 180° turn about its center.


A conjugate axis of length 2b, corresponding to the minor axis of an ellipse, is sometimes drawn on the non-transverse principal axis; its endpoints ±b lie on the minor axis at the height of the asymptotes over/under the hyperbola's vertices. Because of the minus sign in some of the formulas below, it is also called the imaginary axis of the hyperbola.

If b = a, the angle 2θ between the asymptotes equals 90° and the hyperbola is said to be rectangular or equilateral. In this special case, the rectangle joining the four points on the asymptotes directly above and below the vertices is a square, since the lengths of its sides 2a = 2b.

If the transverse axis of any hyperbola is aligned with the x-axis of a Cartesian coordinate system and is centered on the origin, the equation of the hyperbola can be written as

(Our solved example in mathguru.com uses this concept)




The above explanation is copied from Wikipedia, the free encyclopedia and is remixed as allowed under the Creative Commons Attribution- ShareAlike 3.0 Unported License.