Definition of hyperbola, what is the definition of hyperbola?

Hyperbola. 1) Definition The absolute value of the difference between the distance from the two fixed points f1,f2 in the plane is equal to the fixed value 2a(0<2a<|f1f2|) of the point.

The ratio of the distance to the fixed point is e(e 1).

2) Geometric properties:

Focus: Vertices:

Axis of symmetry: x-axis, y-axis.

Eccentricity: The larger the e, the wider the opening.

Alignment: Asymptote:

Focal radius: The line segment connecting any point m on the hyperbola with the focal point of the hyperbola is called the focal radius of the hyperbola.

The focal radius formula for the hyperbola with the focus on the x-axis:

The focal radius formula for hyperbola with a focus on the y-axis:

where are the lower and upper focus of the hyperbola).

Add left and subtract right, add and subtract below", which is the opposite of the parabolic note, and the same as the ellipse note, but with more absolute values).

Focus chord: The intersecting chord formed by the recant hyperbola of the focal point

Diameter: Intersecting chord that is over focus and perpendicular to the axis of symmetry The focal chord formula is applied directly

3) When a=b? Eccentricity e= ?The two asymptotes are perpendicular to each other, respectively, and the hyperbola is an equiaxed hyperbola, which can be set toAt 0, the focus is on the x-axis, and at 0, the focus is on the y-axis.

4) Conjugate hyperbola: The real axis of the known hyperbola is the imaginary axis, and the imaginary axis is the real axis, and the resulting hyperbola is called the conjugate hyperbola of the original hyperbola

Features: Common pair of asymptotics;

The focal point of the original hyperbola and its conjugate hyperbola is on the same circle;

To find the conjugate hyperbola method, change 1 to —1

5) Hyperbola of the asymptotic system: (0, each real value corresponds to a hyperbola).

6) The relationship between the equation of the hyperbola and the asymptotic equation.

If the hyperbolic equation is an asymptote equation:

If the asymptote equation is hyperbola, it can be set to

If the hyperbola has a common asymptote with , it can be set to ( focus on the x-axis, focus on the y-axis).

In general, hyperbola, which literally means "exceeding" or "exceeding", is a class of conic curves defined as two halves of a conical surface with a plane intersecting right angles.

In mathematics, a hyperbola (multiple hyperbola or hyperbola) is a type of smooth curve that lies in a plane and is defined by an equation of its geometric properties or a combination of its solutions. Hyperbolas have two pieces called connected components or branches, which are mirror images of each other, similar to two infinity bows.

A hyperbola is one of three conical sections formed by the intersection of a plane and a bipyramidal. (The other conic parts are parabolas and ellipses, and circles are special cases of ellipses) If the plane intersects the two halves of the cone, but does not pass through the vertices of the cone, the conic curve is hyperbola.

1. Definition of hyperbola: In general, hyperbola is a type of conic curve defined as two halves of a conical surface with a plane intersection of right angles. It can also be defined as the trajectory of a point where the difference in distance from two fixed points (called focal points) is constant.

2. Hyperbola branches: Hyperbola has two branches. When the focus is on the x-axis, it is the left branch and the right branch; When the focus is on the y-axis, it is the upper branch and the lower branch.

3. The vertices of the hyperbola: The hyperbola and the line where its focal line is connected have two intersection points, which are called the vertices of the hyperbola.

4. The real axis of the hyperbola: the line segment between the two vertices is called the real axis of the hyperbola, and half of the length of the real axis is called the semi-solid axis.

5. Asymptote of hyperbola: Hyperbola has two asymptotic lines. Asymptotic and hyperbola do not intersect. The equation for the asymptote is to change the constant on the right side of the standard equation to 0, and the solution of the asymptote can be found by solving the binary quadratic.

Defined as the trajectory of a point where the difference in distance from two fixed points (called focal points) is constant. This fixed distance difference is twice as much as a, where a is the distance from the center of the hyperbola to the vertex of the nearest branch of the hyperbola.

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

The value range of hyperbola:

x a (focus on the x-axis) or y a (focus on the y-axis).

Symmetry of hyperbola:

With respect to the axis and origin symmetry, where the origin is central symmetry.

Vertices of hyperbolas:

a（－a，0），a'（a，0）。At the same time AA'It is called the real axis of the hyperbola and aa'│＝2a。

b（0，－b），b'（0，b）。At the same time bb'It is called the imaginary axis of the hyperbola and bb'│＝2b。

f1 (c,0) or (0, c), f2 (c,0) or (0,c). f1 is the left focus of the hyperbola, f2 is the right focus of the hyperbola and f1f2 2c.

For the real axis, the imaginary axis, and the focal point are: a2+b2=c2.

Hyperbola is defined as a type of conic curve that is two halves of a plane intersection with a right-angled conic surface.

Mathematically Definition: We take the absolute value of the difference between the distance between the two fixed points f1 and f2 in the plane equal to a constant (the constant is 2a, less than |f1f2|) is called hyperbola. i.e.: |pf1|-|pf2│|=2a。

Definition 1: The trajectory of a point in the plane where the absolute value of the difference between the distance to two fixed points is constant (less than the distance between these two fixed points) is called hyperbola. The fixed point is called the focal point of the hyperbola.

Definition 2: The trajectory of a point in the plane where the ratio of the distance to a given point and straight line is constant e((e>1), which is the eccentricity of the hyperbola) is called hyperbola. The fixed point is called the focus of the hyperbola, and the fixed line is called the alignment of the hyperbola.

The equation for hyperbolic alignments is (focus on the x-axis) or (focus on the y-axis).

Definition 3: A plane truncates a conical surface, when the section is not parallel to the bus bar of the conic plane and does not pass through the vertices of the cone, and intersects both cones of the conical surface, the intersecting line is called hyperbola.

Definition 4: In a planar Cartesian coordinate system, the image of the binary quadratic equation f(x,y)=ax2+bxy+cy2+dx+ey+f=0 satisfies the following conditions: 1, a, b, and c are not all zero; 2、δ=b2-4ac>0。

Summary. Hello <>

Hyperbola is a class of classical functional images, which is defined as a graph composed of all points $(x,y)$ satisfying $x 2 a 2 - y 2 b 2 = 1$. where $a$ and $b$ are constants, and $a>b>0$. The hyperbolic image has two branches, showing symmetry.

In a coordinate system, the axis of symmetry of the hyperbola is $y=0$, which is called the real axis; Whereas, the straight line $x=0$ perpendicular to the real axis is called the dashed axis. The two branches of the hyperbola are tangent to the origin with both the real and dashed axes.

Definition of hyperbola.

Hello <>

Hyperbola is a kind of classic function image of Douqin, which defines the hungry hall as a graph composed of all the points $(x,y)$ satisfying $x 2 a 2 - y 2 b 2 = 1$. where $a$ and $b$ are constants, and $a>b>0$. The hyperbolic image has two branches, showing symmetry.

In a coordinate system, the axis of symmetry of the hyperbola is $y=0$, which is called the real axis; The straight line $x=0$ perpendicular to the real axis of the empty limb is called the imaginary axis. The two branches of the hyperbola are tangent to the origin with both the real and dashed axes.

In addition, hyperbola has many important properties and applications:1The equation for hyperbola in polar coordinate system is $r 2=a 2 sec 2 theta-b 2 sin 2 theta$.

2.The asymptote of the hyperbola is $y= pm b a cdot x$. 3.

Hyperbola has a wide range of applications in analytic geometry, such as hyperbolic coordinate system, eccentricity and other concepts. 4.Hyperbola also has important applications in physics, such as propagation in electromagnetic fields and refraction in trapped optics.

Another 1The shape and properties of hyperbolas are very different from ellipses and parabolas, and they have important applications in geometry, algebra, and physics. 2.

Hyperbola is an important type of quadratic curve, and its equation has a standard form, which can be deformed by turning through translation, rotation, etc. 3.In calculus, hyperbolic functions are also widely used, such as hyperbolic sine function $ sinh x$ and hyperbolic cosine function $ cosh x$, etc.

These functions have a wide range of applications in computer science, economics, statistics, and other fields.

Summary. Affinity, definition of hyperbola: the absolute value of the difference between the distance in the plane and the fixed point f1 and f2 is equal to the constant (less than |f1f2|The trajectories of the points are called hyperbolas, the two fixed points are called the focal points of the hyperbola, and the distance between the two focal points is called the focal length of the hyperbola.

Affinity, definition of hyperbola: The absolute value of the difference between the distance between the plane and the definite limb points f1 and f2 is equal to the constant Li Xiao (less than |f1f2|The trajectories of the points are called hyperbolic dots, these two fixed points are called hyperbolic focal points, and the distance between the two focal points is called the focal length of hyperbola.

This question examines mathematical knowledge.

Mathematical absoluteness is a general means for human beings to strictly describe and deduce the abstract structure and pattern of things, which can be applied to any problem in the real world. In this sense, numerical modulosis belongs to the formal sciences, not the natural sciences. Different mathematicians and philosophers have a range of opinions on the exact scope and definition of mathematics.