What Is a Hyperbola?

Generally, a hyperbola (the Greek " ", which literally means "exceeds" or "exceeds") is a type of conic curve defined as the two halves of a plane intercepting a right-angle conical surface.

In mathematics, a hyperbola (multiple hyperbola or hyperbola) is a smooth curve lying in a plane, defined by an equation of its geometric properties or a combination of its solutions. Hyperbola has two pieces, called connected components or branches, which are mirror images of each other, similar to two infinite bows. A hyperbola is one of three conical sections formed by the intersection of a plane and a double cone. (The other conical parts are parabola and ellipse, and the circle is a special case of ellipse.) If the plane intersects the two halves of a double cone, but does not pass through the apex of the cone, the conic curve is a hyperbola.

Hyperbola 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;

The shape as an open orbit (different from a closed elliptical orbit), such as the orbit of a spacecraft during the gravity-assisted swing of the planet, or more generally, any spacecraft exceeding the escape velocity of the nearest planet;

As a path for a single comet (a travel too fast to return to the solar system);

Scattering trajectories as subatomic particles (repulsive rather than attractive, but the principle is the same);

In radio navigation, when the distance to two points can be determined rather than the distance itself, and so on.

Each branch of the hyperbola has two straighter (lower curvature) arms that extend further from the center of the hyperbola. The diagonally opposite arms, one from each branch, tend to a common line, called the asymptote of the two arms. So there are two asymptotes, the intersection of which lies at the center of symmetry of the hyperbola, which can be thought of as a mirror point where each branch reflects to form another branch. In the case of the curve {\ displaystylef (x) = 1 / x} f (x) = 1 / x, the asymptote is two coordinate axes.

Hyperbola sharing many

We put the distance between the two fixed points F ₁ and F ₂ in the plane

The application of hyperbola in practice is a ventilation tower,

If F ₁ PF ₂ = ,

Then S F ₁ PF ₂ = b ² × cot or S F ₁ PF ₂ =

.

Example: It is known that F1 and F2 are hyperbola C : the left and right focal points of x ² y ² = 1, point P is on C , F ₁ PF ₂ = 60 °, then the distance from P to the x axis is more

less?

Solution: From the formula of the hyperbolic focal triangle area:

S F ₁ PF ₂ = b ² × cot =

Let the distance from P to the x-axis be h, then S F ₁ PF ₂ =; h =

Parametric equation.

After the light emitted from one focal point of the hyperbola is reflected by the hyperbola, the reverse extension lines of the reflected light are converged to the other focal point of the hyperbola. The inverse virtual focusing property of hyperbola can also find practical applications in the design of astronomical telescopes.

Optical properties of hyperbola