Let x=0 in the standard equation and get y? =-b？ This equation has no real root. For drawing convenience, draw B 1(0, b) and B2(0, -b) on the Y axis, and take B 1B2 as the imaginary axis.

General, hyperbolic (Greek)? περβολ? ",which literally means" beyond "or" beyond ") is a conic curve defined as two halves of a right-angled cone where planes intersect.

It can also be defined as the trajectory of a point whose distance difference from two fixed points (called focus) is constant. This fixed distance difference is twice that of A, where A is the distance from the center of hyperbola to the vertex of the nearest branch of hyperbola.

A is also called the real semi-axis of hyperbola. The focal point is located on the through axis, and the middle point is called the center, which is generally located at the origin.

Extended data:

Mathematically, hyperbola (multiple hyperbola or hyperbola) is a smooth curve on a plane, which is defined by the equation of its geometric characteristics or the combination of its solutions.

A hyperbola has two parts, called connected components or branches, which are mirror images of each other, similar to two infinite bows. Hyperbola is one of the three conic curves formed by the intersection of plane and double cone.

If the plane intersects the two halves of a double cone, but does not pass through the vertex of the cone, the conic curve is a hyperbola.

Exaggeration appears in many ways:

As a curve representing the function {\ displaystylef (x) =1/x} f (x) =1/x in the Cartesian plane;

As a path for future shadows;

As the shape of an open orbit (different from a closed elliptical orbit), such as the orbit of an spacecraft during the planetary gravity-assisted swing, or more generally, any spacecraft that exceeds the escape speed of the nearest planet;

As the path of a single comet (a comet that runs too fast to return to the solar system);

As the scattering trajectory of subatomic particles (repulsion rather than gravity, but the principle is the same);

In radio navigation, when the distance can be determined from the distance between two points instead of the distance itself, and so on.

Each branch of the hyperbola has two arms, which further extend straighter (lower curvature) from the center of the hyperbola. Diagonally opposite arms, one for each branch, tend to have the same line, which is called the asymptote of these two arms.

So there are two asymptotes, and their intersection points are located in the symmetrical center of hyperbola, which can be regarded as the mirror image points of each branch reflecting to form the other branch. When the curve {\ displaystylef (x) =1/x} f (x) =1/x, the asymptote has two coordinate axes.

Hyperbola * * * enjoys many analytical properties of ellipse, such as eccentricity, focus and pattern. Many other mathematical objects are related to hyperbola, such as hyperbolic paraboloid, hyperboloid, hyperbolic geometry, hyperbolic function, gyro vector space and so on.

References:

Baidu encyclopedia-hyperbola