A geometric formation, which is called a hyperbola, is a plane curve 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 the 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 point of point construction of a hyperbole, this is the sum of the points on the plane. Each such point is characterized by a constant modulus of the difference in the distance from the two focal centers.
A flat curve is distinguished by the main features that are inherent only to it:
- Hyperbolas are two separate lines called branches.
- In the middle of the axis of a large order is the center of the figure.
- A vertex is the point of two branches closest to each other.
- Focal distance denotes the distance from the center of the curve to one of the foci (indicated by the letter "c").
- The major axis of the hyperbola describes the shortest distance between the branch-lines.
- The tricks lie on a major axis under the condition that the distance from the center of the curve is the same. The line that supports the major axis is called the transverse 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").
- 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 the formulas under the letter "b".
In classical Cartesian coordinates, the well-known equation by which the construction of a hyperbola is possible looks like this: (x 2 / a 2 ) - (y 2 / b 2 ) = 1. That type of curve that has the same semi-axes is called isosceles. In a rectangular coordinate system, it can be described by a simple equation: xy = a 2/2, and the foci of the hyperbola must be located at the intersection points (a, a) and (−a, −a).
A parallel hyperbole may exist for each curve. This is its conjugate version, in which the axes are interchanged, and the asymptotes remain in place. The optical property of the figure is that the light from an imaginary source in one focus is able to be reflected by the second branch and intersect in 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 through 180 ° in the center.
The eccentricity of the hyperbola is determined by the numerical characteristic of the conical section, which shows the degree of deviation of the section from the ideal circle. In mathematical formulas, this indicator is indicated by the letter "e". 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.