We can give definitions of hyperbola in different ways.
This definition may be expressed also in terms of tangent circles. The center of any circles externally tangent to two given circles lies on a hyperbola, whose foci are the centers of the given circles and where the vertex distance 2a equals the difference in radii of the two circles. As a special case, one given circle may be a point located at one focus; since a point may be considered as a circle of zero radius, the other given circle—which is centered on the other focus—must have radius 2a. This provides a simple technique for constructing a hyperbola, as shown below. It follows from this definition that a tangent line to the hyperbola at a point P bisects the angle formed with the two foci, i.e., the angle F1P F2. Consequently, the feet of perpendiculars drawn from each focus to such a tangent line lies on a circle of radius a that is centered on the hyperbola's own center.
The eccentricity of the conic section obtained by reciprocation is the ratio of the distances between the two circles' centers to the radius r of reciprocation circle C. If B and C represent the points at the centers of the corresponding circles, then
This definition implies that the hyperbola is both the locus of the poles of the tangent lines to the circle B, as well as the envelope of the polar lines of the points on B. Conversely, the circle B is the envelope of polars of points on the hyperbola, and the locus of poles of tangent lines to the hyperbola. Two tangent lines to B have no (finite) poles because they pass through the center C of the reciprocation circle C; the polars of the corresponding tangent points on B are the asymptotes of the hyperbola. The two branches of the hyperbola correspond to the two parts of the circle B that are separated by these tangent points.
Conic section
A hyperbola may be defined as the curve of intersection between a right circular conical surface and a plane that cuts through both halves of the cone. The other major types of conic sections are the ellipse and the parabola; in these cases, the plane cuts through only one half of the double cone. If the plane is parallel to the axis of the double cone and passes through its central apex, a degenerate hyperbola results that is simply two straight lines that cross at the apex point.
Difference of distances to foci
A hyperbola may be defined equivalently as the locus of points where the absolute value of the difference of the distances to the two foci is a constant equal to 2a, the distance between its two vertices. This definition accounts for many of the hyperbola's applications, such as trilateration; this is the problem of determining position from the difference in arrival times of synchronized signals, as in GPS.
This definition may be expressed also in terms of tangent circles. The center of any circles externally tangent to two given circles lies on a hyperbola, whose foci are the centers of the given circles and where the vertex distance 2a equals the difference in radii of the two circles. As a special case, one given circle may be a point located at one focus; since a point may be considered as a circle of zero radius, the other given circle—which is centered on the other focus—must have radius 2a. This provides a simple technique for constructing a hyperbola, as shown below. It follows from this definition that a tangent line to the hyperbola at a point P bisects the angle formed with the two foci, i.e., the angle F1P F2. Consequently, the feet of perpendiculars drawn from each focus to such a tangent line lies on a circle of radius a that is centered on the hyperbola's own center.
A proof that this characterization of the hyperbola is equivalent to the conic-section characterization can be done without coordinate geometry by means of Dandelin spheres.
Directrix and focus
A hyperbola can be defined as the locus of points for which the ratio of the distances to one focus and to a line (called the directrix) is a constant 
that is larger than 1. This constant is the eccentricity of the hyperbola. By symmetry a hyperbola has two directrices, which are parallel to the conjugate axis and are between it and the tangent to the hyperbola at a vertex.
that is larger than 1. This constant is the eccentricity of the hyperbola. By symmetry a hyperbola has two directrices, which are parallel to the conjugate axis and are between it and the tangent to the hyperbola at a vertex.Reciprocation of a circle
The reciprocation of a circle B in a circle C always yields a conic section such as a hyperbola. The process of "reciprocation in a circleC" consists of replacing every line and point in a geometrical figure with their corresponding pole and polar, respectively. The pole of a line is the inversion of its closest point to the circle C, whereas the polar of a point is the converse, namely, a line whose closest point to C is the inversion of the point.
The eccentricity of the conic section obtained by reciprocation is the ratio of the distances between the two circles' centers to the radius r of reciprocation circle C. If B and C represent the points at the centers of the corresponding circles, then
Since the eccentricity of a hyperbola is always greater than one, the center B must lie outside of the reciprocating circle C.
This definition implies that the hyperbola is both the locus of the poles of the tangent lines to the circle B, as well as the envelope of the polar lines of the points on B. Conversely, the circle B is the envelope of polars of points on the hyperbola, and the locus of poles of tangent lines to the hyperbola. Two tangent lines to B have no (finite) poles because they pass through the center C of the reciprocation circle C; the polars of the corresponding tangent points on B are the asymptotes of the hyperbola. The two branches of the hyperbola correspond to the two parts of the circle B that are separated by these tangent points.
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