Geometric formation, which is called a hyperbola, is a flat curve figure of the second order, consisting of two curves that are drawn separately and do not intersect. The mathematical formula for its description looks like this: y=k/x, if the number under index k is not equal to zero. In other words, the vertices of the curve constantly tend to zero, but will never intersect with it. From the standpoint of point construction, a hyperbola is the sum of points on a plane. Each such point is characterized by a constant value of the modulus of the difference between the distance from two focal centers.
A flat curve is distinguished by the main features that are unique to it:
- A hyperbola is two separate lines called branches.
- The center of the figure is located in the middle of the high order axis.
- A vertex is a point of two branches closest to each other.
- Focal distance refers to the distance from the center of the curve to one of the foci (denoted by the letter "c").
- The major axis of the hyperbola describes the shortest distance between branches-lines.
- Focuses lie on the major axis provided the same distance from the center of the curve. The line that supports the major axis is calledtransverse axis.
- The semi-major axis is the estimated distance from the center of the curve to one of the vertices (indicated by the letter "a").
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building a hyperbola A straight line passing perpendicular to the transverse axis through its center is called the conjugate axis.
- The focal parameter determines the segment between the focus and the hyperbola, perpendicular to its transverse axis.
- The distance between the focus and the asymptote is called the impact parameter and is usually encoded in formulas under the letter "b".
In classical Cartesian coordinates, the well-known equation that makes it possible to construct a hyperbola looks like this: (x2/a2) – (y 2/b2)=1. The type of curve that has the same semiaxes is called isosceles. In a rectangular coordinate system, it can be described by a simple equation: xy=a2/2, and the hyperbola foci should be located at the intersection points (a, a) and (−a, −a).
To each curve there can be a parallel hyperbola. This is its conjugate version, in which the axes are reversed, and the asymptotes remain in place. The optical property of the figure is that light from an imaginary source at one focus is able to be reflected by the second branch and intersect at the second focus. Any point of a potential hyperbola has a constant ratio of the distance to any focus to the distance to the directrix. A typical plane curve can exhibit both mirror and rotational symmetry when rotated 180° through the center.
The eccentricity of the hyperbola is determined by the numerical characteristic of the conic section, which shows the degree of deviation of the section from the ideal circle. In mathematical formulas, this indicator is denoted by the letter "e". The eccentricity is usually invariant with respect to the motion of the plane and the process of transformations of its similarity. A hyperbola is a figure in which the eccentricity is always equal to the ratio between the focal length and the major axis.
