The conjugate hyperbola is given by

The hyperbola and conjugate hyperbola are separated by two diagonal asymptotes which form the set of null elements:

Given a hyperbola with asymptote A, its reflection in A produces the conjugate hyperbola.

Any diameter of the original hyperbola is reflected to a conjugate diameter.

As E. T. Whittaker wrote in 1910, " hyperbola is unaltered when any pair of conjugate diameters are taken as new axes, and a new unit of length is taken proportional to the length of either of these diameters.

For example, the isogonal conjugate of a line is a circumconic ; specifically, an ellipse, parabola, or hyperbola according as the line intersects the circumcircle in 0, 1, or 2 points.

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.

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 F < sub > 1 </ sub > and F < sub > 2 </ sub >, and the thin black line joining them is the transverse 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.

Consistent with the symmetry of the hyperbola, if the transverse axis is aligned with the x-axis of a Cartesian coordinate system, the slopes of the asymptotes are equal in magnitude but opposite in sign, ±, where b = a × tan ( θ ) and where θ is the angle between the transverse axis and either asymptote.

Because of the minus sign in some of the formulas below, it is also called the imaginary axis of the hyperbola.

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

Likewise, a hyperbola with its transverse axis aligned with the y-axis is called a " North-South opening hyperbola " and has equation

A hyperboloid of revolution of one sheet can be obtained by revolving a hyperbola around its semi-minor axis.

A hyperboloid of revolution of two sheets can be obtained by revolving a hyperbola around its semi-major axis.

For example, the pencil of curves ( 1-dimensional linear system of conics ) defined by is non-degenerate for but is degenerate for concretely, it is an ellipse for two parallel lines for and a hyperbola with < math > a < 0 </ math > – throughout, one axis has length 2 and the other has length which is infinity for

Either half of the minor axis is called the semi-minor axis, of length b. Denoting the semi-major axis length ( distance from the center to a vertex ) as a, the semi-minor and semi-major axes ' lengths appear in the equation of the hyperbola relative to these axes as follows:

: is length of semi-major axis of orbit's hyperbola.

For comparison, the other two general conic sections, the ellipse and the parabola, derive from the corresponding Greek words for " deficient " and " comparable "; these terms may refer to the eccentricity of these curves, which is greater than one ( hyperbola ), less than one ( ellipse ) and exactly one ( parabola ), respectively.

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 ).

A hyperbola meets it at two real points corresponding to the two directions of the asymptotes.

If the conic is a parabola then the inverse will be a cardioid, if the conic is a hyperbola then the corresponding limaçon will have an inner loop, and if the conic is an ellipse then the corresponding limaçon will have no loop.

so negative K corresponds to an ellipse and positive K to a hyperbola, with the rectangular case of the squeeze mapping corresponding to K = 1.

A parabola is an ellipse that is tangent to the line at infinity Ω, and the hyperbola is an ellipse that crosses Ω.

Similar to a parabola, a hyperbola is an open curve, meaning that it continues indefinitely to infinity, rather than closing on itself as an ellipse does.

Orbits that form a hyperbola or an ellipse are much more common.

* In conic sections, it is said of two ellipses, two hyperbolas, or an ellipse and a hyperbola which share both foci with each other.

If an ellipse and a hyperbola are confocal, they are perpendicular to each other.

A central conic is called an ellipse or a hyperbola according as it has no asymptote or two asymptotes.

* Polar distance ( astronomy ), an astronomical term associated with the celestial equatorial coordinate system Σ ( α, δ ) ellipse and lower, a hyperbola

There are clear advantages in being able to think of a hyperbola and an ellipse as distinguished only by the way the hyperbola lies across the line at infinity ; and that a parabola is distinguished only by being tangent to the same line.

It was Apollonius who gave the ellipse, the parabola, and the hyperbola the names by which we know them.

It is the form of the fundamental property ( expressed in terms of the " application of areas ") that leads him to give these curves their names: parabola, ellipse, and hyperbola.

Note that over the complex numbers factors as and is degenerate because reducible, while defines a non-degenerate curve, an ellipse / hyperbola ( these are not distinct over the complex numbers, because there is no sense of positive or negative ).

# a lemma upon the Surface Loci of Euclid which states that the locus of a point such that its distance from a given point bears a constant ratio to its distance from a given straight line is a conic, and is followed by proofs that the conic is a parabola, ellipse, or hyperbola according as the constant ratio is equal to, less than or greater than 1 ( the first recorded proofs of the properties, which do not appear in Apollonius ).

In projective geometry, Pascal's theorem ( aka Hexagrammum Mysticum Theorem ) states that if an arbitrary six points are chosen on a conic ( i. e., ellipse, parabola or hyperbola ) and joined by line segments in any order to form a hexagon, then the three pairs of opposite sides of the hexagon ( extended if necessary ) meet in three points which lie on a straight line, called the Pascal line of the hexagon.

A cylinder whose cross section is an ellipse, parabola, or hyperbola is called an elliptic cylinder, parabolic cylinder, or hyperbolic cylinder respectively.

It is one of the axes of symmetry for the curve: in an ellipse, the shorter one ; in a hyperbola, the one that does not intersect the hyperbola.

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.

There are therefore two asymptotes, whose intersection is at the center of symmetry of the hyperbola, which can be thought of as the mirror point about which each branch reflects to form the other branch.

Every hyperbola is congruent to the origin-centered East-West opening hyperbola sharing its same eccentricity ε ( its shape, or degree of " spread "), and is also congruent to the origin-centered North-South opening hyperbola with identical eccentricity ε — that is, it can be rotated so that it opens in the desired direction and can be translated ( rigidly moved in the plane ) so that it is centered at the origin.

These solutions yield good rational approximations of the form x / y to the square root of n. In Cartesian coordinates, the equation has the form of a hyperbola ; it can be seen that solutions occur where the curve has integral ( x, y ) coordinates.

Any conic that passes through the four orthocentric points can only be a rectangular hyperbola.

* A hyperbola can degenerate into two lines crossing at a point, through a family of hyperbolas having those lines as common asymptotes.

This curve can be obtained as the inverse transform of a hyperbola, with the inversion circle centered at the center of the hyperbola ( bisector of its two foci ).

Note that in a hyperbola b can be larger than a.

For example, foci can be used in defining conic sections, the four types of which are the circle, ellipse, parabola, and hyperbola.

A hyperbola can be defined as the locus of points for each of which the absolute value of the difference between the distances to two given foci is a constant.

The same construction can also be applied to the hyperbola.

It is called hyperbolic motion because the equation describing the path of the object through spacetime is a hyperbola, as can be seen when graphed on a Minkowski diagram.