What is the third definition of hyperbola and what is the third definition of hyperbola?

Proof: Equiaxed hyperbola.

The equation is: x 2 a 2-y 2 a 2 = 1, that is, x 2-y 2 = a 2 = k, k is a constant, the two asymptotic line equations are x+y=0 and x-y=0 respectively, let any point on the hyperbola m(x0, y0), the distance from the point m to the two asymptotic lines is:

d1=|x0+y0|/sqrt(2),d2=|x0-y0|sqrt(2), then, d1 d2=(x0 2-y0 2) 2, and x0,y0 satisfies the hyperbolic equation.

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.

HyperbolaThe third definition is that the product of the slope of the change from the moving point in the plane to the two fixed points a1(a,0) and a2(-a,0) is equal to the constant e 2-1The trajectory of the pointIt's called an ellipse or hyperbola. Two of these points are the vertices of an ellipse or hyperbola. When the constant is greater than -1 and less than 0, it is an ellipse; When the constant is greater than 0, it is hyperbola.

The difference in distance from two fixed points (called focal points) is the trajectory of a constant point, and this fixed distance difference is twice that of a.

The third definition of the nature of the curve

The trajectory of the point where the product of the slope of the in-plane moving point to the two fixed points a1(a,0) and a2(-a,0) is equal to the constant e-1 is elliptical or hyperbola. Two of these points are the vertices of elliptic circles or hyperbolas. When 01 is hyperbola.

Conic. The (incomplete) uniform definition of quadratic curves is the trajectory of a point where the quotient of the distance to the fixed point (focal point) and the distance to the fixed line (quasi-line) is the constant e (eccentricity). When e>1, it is a hyperbola, and when e=1, it is a parabola.

When 0 is parallel to the bus of the secondary cone, but not the vertex of the cone, the result is a parabola. When the plane is parallel to the bus of the quadratic cone and passes the vertex of the cone, the result is degenerated into a straight line.

The third definition of hyperbola: x 2-y 2=a 2=k, hyperbola is one of the three types of conical sections formed by the intersection of a plane and a bipyramidal. 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.

It can also be defined as the trajectory of a point where the difference in distance from two fixed 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, which 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.

Summary of the basic knowledge points of hyperbola:

We take the absolute value of the difference between the distance between the two fixed points f1 and f2 in the plane.

A trajectory equal to one constant (constant 2a) is called hyperbola. The absolute value of the distance difference from the plane to the two fixed points is the fixed length.

The trajectory of the point is called hyperbola) i.e.: pf1 - pf2 = 2a. 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 1) is called hyperbola.

The fixed point is called the focus of the hyperbolic dust fingerline. The trajectory of a point in the plane where the ratio of the distance to a given point and the distance to the straight line is a constant greater than 1 is called hyperbola. The fixed point is called the focus of the hyperbola, and the fixed line is called the alignment of the hyperbola.

When the cross-section is not parallel to the bus bar of the conical surface, and both cones of the conical surface intersect, the intersecting line is called hyperbola. Rubber masses in a planar Cartesian coordinate system.

, binary quadratic equations.

f(x,y)=ax 2+bxy+cy 2+dx+ey+f=0 The image is hyperbolic if the following conditions are met.

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.

Question 1: Finding the third definition of conic curve and how to understand it? Definition: The trajectory of a point from a moving point in the plane to a point where the product of the slope of the two fixed points a1(a,0) and a2(-a,0) is equal to the constant e2-1 is an ellipse or hyperbola.

Two of these points are vertices of an ellipse or hyperbola.

When 01 is hyperbola.

Conic curve: Use a plane to cut a conic surface, and the resulting intersection line is called a conic section.

Conic curves are commonly referred to as ellipses, hyperbolas, and parabolas, but strictly speaking, they also include some degradation cases. Specifically:

1) When the plane is parallel to the bus of the conic and does not exceed the vertex of the cone, the result is a parabola.

2) When the plane is parallel to the bus of the conic and passes the vertex of the cone, the result degenerates into a straight line.

3) When the plane intersects only one side of the cone, and does not exceed the vertices of the cone, the result is an ellipse.

4) When the plane intersects only one side of the cone, and does not exceed the vertex of the cone, and is perpendicular to the axis of symmetry of the cone, the result is a circle.

5) When the plane intersects only one side of the cone, passes the vertex of the cone, and is perpendicular to the axis of symmetry of the cone, the result is a point.

6) When both sides of the plane and the cone intersect, and the macro is not more than the vertex of the cone, the result is one of the hyperbolas (the other is the intersection of the top cone of the conic and the plane).

7) When both sides of the plane and the conic intersect and pass the vertices of the cone, the result is two intersecting straight lines.

Reference: Problem 2: Find the second definition of ellipse and hyperbola! The second definition of ellipse and hyperbola is the definition of parabola. This is actually a uniform definition of a cone ding.

Definition: The trajectory of a point where the ratio of the distance to the fixed point to the distance to the fixed line is a constant (e) is a conic curve.

e (0,1) is an ellipse;

When e=1, it is a spring book parabola;

e (1,+) is hyperbola.

A fixed line is the corresponding alignment.

Question 3: What does the left branch of the hyperbola mean y=k x, when k 0, the hyperbola is in the first and third quadrants, then x 0, the hyperbola is in the first quadrant (right); x 0, hyperbola in the third quadrant (left).

When k 0 and the hyperbola is in the second quadrant, then x 0 and the hyperbola is in the second quadrant (left); x 0, hyperbola in quadrant 4 (right).

Problem 4: If (2,k) is a point on a hyperbola, then the image of the function passes through ( )a.

1. Three-quadrant b.

2. Four-quadrant c b analysis: first substitute (2,k) into the hyperbola y= to find the value of k, and then substitute the value of k into the function y=(k-1)x to find the analytical formula of this function, and then solve it according to the characteristics of the proportional function Solution: substitute (2,k) into the hyperbola y= to get, k= , substitute k= into the function y=(k-1)x, y=- x, so the image of the function has passed.

2. Quadrant B Comment: The law used in this problem: in the straight line y=kx, when k 0, the function is imaged.

1. Three quadrants; When k 0, the function is imaged too.

2. Four quadrants