1, definition

In a plane, the locus (or set) of points with equal distances to a fixed point F and a fixed line L is called a parabola. In addition, f is called "the focus of parabola" and l is called "the directrix of parabola".

Define the distance from the focal point to the parabolic directrix as "focal length" and use p>0.

Insert the tangent plane into a cone parallel to the ground to get a circle. If you tilt the plane parallel to one side, you can make a parabola.

2. The standard equation of parabola

Parabola with right opening: y 2 = 2px

Left open parabola: y 2 =-2px

Upper parabola: y = x 2/2p

Parabola of lower opening: y =-x 2/2p

3. Parabola related parameters (for parabola with opening to the right)

Eccentricity: e= 1

Focus: (P/2,0)

Alignment equation l:x=-p/2

Vertex: (0,0)

4. Its analytical solution:

Three-point substitution method

5. Optical properties of parabola;

The light passing through the focus is parallel to the axis of symmetry of the parabola after being reflected by the parabola.

6. Others

Parabola: y = ax*+bx+c

Y equals ax plus bx plus c squared.

When a> is 0, the opening is upward.

When a< is 0, the opening is downward.

When c = 0, the parabola passes through the origin.

When b = 0, the axis of symmetry of parabola is the Y axis.

And the vertex y = a (x-h) *+k.

That is, y equals a times the square of (x-h)+K.

H is x of vertex coordinates.

K is y of vertex coordinates.

Generally used to find the maximum and minimum.

Parabolic standard equation: y 2 = 2px

It means that the focus of parabola is on the positive semi-axis of X, the focal coordinate is (p/2,0), and the directrix equation is x=-p/2.

Since the focus of parabola can be on any semi-axis, * * has the standard equation y 2 = 2px y 2 =-2px x 2 = 2py x 2 =-2py.

Ellipse:

definition

An ellipse is a conic curve (also called conic curve). Now there are two definitions in high school textbooks:

1, a point set whose sum of the distances between two points on the plane is a fixed value (the fixed value is greater than the distance between the two points) (these two fixed points are also called the focus of an ellipse, and the distance between the focuses is called the focal length);

2. The ratio of the distance from the plane to the fixed point to the distance from the fixed line is a constant point set (the fixed point is not on the fixed line, and the constant is a positive number less than 1) (the fixed point is the focus of the ellipse, and the straight line is called the directrix of the ellipse). These two definitions are equivalent.

Standard equation

In the plane rectangular coordinate system, high school textbooks describe ellipses with equations. The standard equation of ellipse is: x 2/a 2+y 2/b 2 =1.

In which a>0 and b>0. The larger one of A and B is the long semi-axis of the ellipse, and the shorter one is the short semi-axis (the ellipse has two symmetry axes and two line segments after being cut by the ellipse, which are called the long semi-axis and the short semi-axis of the ellipse respectively) when A >; B, the focus is on the X axis, the focal length is 2 * (A 2-B 2) 0.5, and the directrix equation is X = A 2/C and X =-A 2/C.

The area of an ellipse is πab. An ellipse can be regarded as the stretching of a circle in a certain direction, and its parameter equations are: x=acosθ, y=bsinθ.

formula

Area formula of ellipse

S=π (pi) ×a×b (where a and b are the lengths of the major axis and minor axis of an ellipse, respectively).

Or S=π (pi) ×A×B/4 (where a and b are the lengths of the major axis and minor axis of the ellipse, respectively).

Perimeter formula of elliptic circle

Ellipse circumference has no formula, just an integer or infinite expansion.

The exact calculation of ellipse circumference (L) requires the summation of integral or infinite series. such as

(0-pi/2) integral of l = 4a * sqrt (1-e sin t), where a is the major axis of the ellipse and e is eccentricity.

Eccentricity formula of ellipse

e=c/a

The directrix equation of ellipse

x=+-a^2/C

Elliptic focal radius formula

The radius of ellipse passing through the right focus is r=a-ex.

The radius of the left focus is r=a+ex.

Correlation characteristic

Because the figure obtained by plane truncated cone (or cylinder) may be ellipse, it belongs to conic curve.

For example, there is a cylinder, which is cut to obtain a cross section. Prove to be an ellipse (with the first definition above):

Squeeze two hemispheres with the same radius as the cylinder from both ends to the middle, and stop when they touch the section. Then you will get two common points, which are obviously the tangent points of the section and the ball.

Let two points be F 1 and F2.

For any point P on the cross section, the generatrix Q 1 and Q2 passing through P are cylinders, and the great circle tangent to the sphere and cylinder intersects at Q 1 and Q2 respectively.

Then PF 1=PQ 1 and PF2=PQ2, so PF 1+PF2=Q 1Q2.

According to the definition of 1, the cross section is an ellipse with f 1 and F2 as the focus.

In the same way, it can also be proved that the oblique section of the cone (which does not pass through the bottom) is elliptical.

Hyperbola:

definition

Mathematically, the trajectory formed when the absolute value of the distance difference between a moving point and two fixed points on the plane is always a certain value is called hyperbola. These two fixed points are called the focus of hyperbola.

The second definition of hyperbola is:

The ratio of the distance to the fixed point to the distance to the fixed line =e, e∈( 1, +∞).

The standard equation of hyperbola is (x 2/a 2)-(y 2/b 2) = 1.

Where a>0, b>0, C2 = A 2+B 2, and the absolute value of the difference between a fixed point and two fixed points is a fixed value 2a.

The parametric equation of hyperbola is:

x=X+a secθ

y=Y+b tanθ

(θ is a parameter)

Geometric properties:

1, value range: x ≥ a, x ≤-a.

2. Symmetry: Symmetry about coordinate axis and origin.

3. Vertex: A(-a, 0) A'(a, 0) AA' is called the real axis of hyperbola, which is 2a long;

B(0, -b) B'(0, b) BB' is called the imaginary axis of hyperbola, and its length is 2b.

4. Asymptote:

y= (b/a)x

5. Centrifugal rate:

E=c/a value range: (1, +∞)

The ratio of the distance between a point on a hyperbola and a fixed point to a fixed straight line is equal to the eccentricity of the hyperbola.

7 hyperbolic focal radius formula: the distance from any point to the focus on the conic.

The real and imaginary axes of equilateral hyperbola are equal in length.

2a=2b e=√2

9 *** Yoke Hyperbola

(x 2/a 2)-(y 2/b 2) = 1 and (y 2/b 2)-(x 2/a 2) = 1 are called equilateral hyperbola.

(1)*** asymptote

(2)e 1+E2 & gt； =2√2

The standard formula of hyperbola is:

x^2/a^2-y^2/b^2 = 1(a & gt； 0，b & gt0)

The standard form of inverse proportional function is xy = c (c

But the inverse proportional function is really rotated by hyperbolic function, and the rotation angle can be set to a (a

(A is the inclination of hyperbolic asymptote)

Then there is

X = xcosa + ysina

Y = xcosa - ysina

x^2-y^2 =(xcosa+ysina)^2-(xcosa-ysina)^2

= 4xy(cosasina)

= 4c(cosasina)

therefore

x^2/4c(cosasina)-y^2/4c(cosasina)= 1(4c(cosa Sina)>0)

y^2/(-4c(cosasina))-x^2/(-4c(cosasina))= 1(4c(cosa Sina)& lt； 0)

It is proved that the inverse proportional function is actually a hyperbolic function.