Regarding the question of the eccentricity of hyperbola, the eccentricity of hyperbola

y 2 a 2-x 2 b 2=1 asymptote is y= ax b, first consider y 2 a 2-x 2 b 2=1 asymptote y=ax b tangent to the parabola y=x 2+1.

The simultaneous y=ax b and y=x 2+1 solution yields: x= 2, i.e., the abscissa of the intersection is x= 2

Derivative of x on both sides of y=x 2+1, y'=2x

Because the asymptotic line of the hyperbola y 2 a 2-x 2 b 2 = 1 (a>0, b>0) is tangent to the parabola y=x 2+1, the slope of y=x 2+1 at the intersection is equal to the slope of y=ax b, i.e., =a b, so that a b=2, a=2b, c 2=a 2+b 2=5b 2, c = 5b

The eccentricity of the hyperbola is e=ca=5b2b=52.

The eccentricity of the hyperbola can also be calculated when y2 a 2-x 2 b 2 = 1 asymptote y=-ax b tangent to the parabola y=x 2+1.

Take, for example, the focus on the x-axis.

a,0) is the vertex of the real axis, that is, the intersection point of the hyperbola and the coordinate axis.

b,0) is the vertex of the dotted axis.

c,0) is the focal coordinate.

Their relationship satisfies a 2 + b 2 = c 2

Eccentricity e=c a

Having this information should be enough to draw conclusions.

I can't draw you here, so it's hard to explain, but you can definitely push it out yourself, it's not that hard.

I feel that the research method of this problem is like the method of controlling variables.

You first find a piece of manuscript paper and draw a picture.

The way to do this is to find a few special values to draw first.

The key to hyperbolic drawing is to draw the asymptotic line, the equation for the asymptote line is y= (b a)x, draw it, and the approximate image of the hyperbola can be determined.

The opening you are talking about getting bigger or smaller is determined by the position of the asymptote, which means that the change in the slope of the asymptote (b a) affects the size of the hyperbolic opening.

However, there does not seem to be an absolute connection between (b a) and eccentricity (c a).

If the hyperbola is on the x-axis: then it is (-a,0)(a,0).

If the hyperbola is on the y-axis: then it is (0,-a)(0,a).

The trajectory of a point in the plane where the absolute value of the difference between the distance to two fixed points is a constant 2a (less than the distance between these two fixed points) is called hyperbola. The fixed point is called the focal point of the hyperbola, and the distance between the two focal points is called the focal length, which is denoted by 2C.

In the plane, the ratio of the distance to a given point and the straight line is a constant e(e>1, which is the eccentricity of the hyperbola; The trajectory of a point where the fixed point is not on a fixed line is called a hyperbola. The fixed point is called the focus of the hyperbola, and the fixed line is called the alignment of the hyperbola.

1. Value area: x a, x -a or y a, y -a2, symmetry: about the coordinate axis and the origin symmetry front.

3. Vertices: a(-a,0) a'(a,0) aa' is called the real axis of the hyperbola, 2 a;

b(0,-b) b'(0,b) bb' is called the imaginary axis of the hyperbola, 2b long.

4. Asymptote:

Transverse axis: y= (b a)x

Vertical axis: Silver Rock y= (a b)x

5. Eccentricity:

e=c a Value range: (1,+.)

6 The ratio of the distance from a point on the hyperbola to the fixed point and the distance to the fixed line (corresponding alignment group) is equal to the eccentricity of the hyperbola.

Summary. Eccentricity of hyperbola: The ratio of the focal length of the hyperbola to the length of the real axis ca frac ac is called the eccentricity of the hyperbola, which is expressed as EEE.

Eccentricity of hyperbola: The ratio of the focal length of the hyperbola to the length of the old shaft is called the centrifugal rate of the hyperbola, which is expressed by EEE.

Eccentricity of hyperbola: The ratio of the focal length of the hyperbola to the length of the real axis is called the eccentricity of the hyperbola, which is expressed as EEE.

The eccentricity formula for hyperbola: e= (a -b) a. where a is the length of the semi-major axis of the ellipse, and b is the length of the semi-minor axis of the ellipse.

In mathematics, hyperbola is a class of conic curves defined as two halves of a plane intersection of right-angled conic surfaces. 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. 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 point is located on the through axis, and the middle point of their rotten filial piety is called the center, and the center is generally located at the origin.

Summary. Hyperbolic eccentricity.

Hyperbolic eccentricity.

The formula for hyperbolic eccentricity is e=c a = a +b ) a = 1+(b a) ] In mathematics, hyperbola is a class of conic curves defined as two halves of a plane high-bending trundly angled conic surface. It can also be defined as the trajectory of a point where the distance from two fixed points is constant. This fixed distance difference is twice as much as a, where a is the Ruhu distance from the center of the hyperbola to the vertex of the nearest branch of the hyperbola.

Algebraically speaking, hyperbola is a curve on the Cartesian plane defined by the following equation such that all coefficients are real slag and there are more than one solution of point pairs (x,y) defined on the hyperbola. Note: On the Cartesian coordinate plane, the image of two reciprocal variables is hyperbola.

1.√2≤e≤2

>2≤c/a≤2

>2≤c^/a^≤4

>2-1≤c^/a^

>1≤b^/a^≤3

>1≤b/a≤√3

When b a=1, the two asymptotic lines of the hyperbola are: y= x, and it is easy to judge that the angle between the two is 90°;

When b a = 3, the asymptote of the hyperbola is: y = 3x, the inclination angles are 60°, 120° respectively, and the angle between the two boxes is 60° (note that the side less than or equal to 90° should be taken, not the obtuse angle!). ）

Therefore, the range of m is [60°, 90°].

2.From the nature of hyperbola, it can be seen that there must be |af1|=|bf1|,|f1f2|⊥|ab|

af2|=|bf2|, to make isosceles abf2 an acute triangle, as long as its apex angle af2b is acute.

And |f1f2|To divide af2b equally, just make af2f1 45°.

i.e.: tan af2f1=|af1|/|f1f2|<1

>af1|<|f1f2|

Obviously, |f1f2|=2c

af1|<2c

According to the definition of hyperbola, it can be seen that: ||af1|-|af2||=2a

As can be seen from the figure: |af1|<|af2|

Yes |af2|=|af1|+2a

In RT AF2F1, the Pythagorean theorem is by:

af2|^=af1|^+f1f2|^

Substitution|af2|=2a+|af1|,|f1f2|=2c, you get:

af1|=(c^-a^)/a

will bring it into the inequality :

c^-a^)/a<2c

Let e=c a, c=ae, and substitute to obtain:

e^-2e-1<0

>e<√2+1

The eccentricity range of hyperbola is: (1, 2+1).

Hyperbola. , c 2 = a 2 + b 2, eccentricity e = c the coordinates of the a>1f are (-c, 0), and the coordinates of the dispersed e are (a, 0) x=-c, substituting the hyperbolic equation to obtain a(-c, b 2 a), b(-c, -b 2 a).

The triangle abe is an acute triangle, then the slope of be: b 2 a (a+c) < 1 so b 2

So 2a-c>0, i.e. c a=e<2

Therefore, the range of the elastometry of the eccentricity of the hyperbola is (1,2).

e=c a=distance from the point to the focal point and the distance from the alignment.

Definition (2): The trajectory of a point in the plane where the ratio of the distance to a given point and the straight line is constant e(e=c a(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 x= a c (focus on the x-axis) or y= a c (focus on the y-axis).

In the hyperbola, C 2 = A 2 + B 2, the eccentricity E = C A>1f coordinates are (-c, 0), and the coordinates of e are (a, 0) Substituting x=-C into the hyperbolic equation to obtain a(-c, b 2 a), b(-c, -b 2 a).

The triangle abe is an acute triangle, then the slope of be: b 2 a (a+c) < 1 so b 2

0 so 2a-c >0, i.e. c a=e<2

Therefore, the value range of the eccentricity e of the hyperbola is (1,2).

The asymptote of the hyperbola is.

y=b/a*x

or y=-b a*x

From the formula for the distance from the point to the line, we know that d = a b divided by the square of the root number a plus the square of b, and the square of the point p and the distance to a = the square of b, which is brought in by the point p and the distance, is e=c a, c = the root number is 2 times a

So the eccentricity is 2