Analytic geometry: Fifth chapter two times curve (1) Circular elliptic hyperbola _ analytic geometry

§1 Circle

1. The equation of the circle

Graphics

Equation

Center

Radius

1º Standard equation:

x²+y²=r²

2º parameter equation:

3º Polar coordinate equation:

Ρ=r

G (0, 0)

R=r

1º (x-a) ²+ (y-b) ²=r²

2º parameter equation:

G (A, B)

R=r

1ºx²+y²+2mx+2ny+q=0

(M²+N²>Q)

G (-m,-n)

1ºx²+y²=2rx

2º Polar coordinate equation:

Ρ=2rcosφ

G (r,0)

R=r

1ºx²+y²=2ry

2º Polar coordinate equation:

Ρ=2rsinφ

G (0,r)

R=r

Polar coordinate equation:

Ρ²-2ρρ0cos (φ-φ0) +ρ02=r2

G (ρ0,φ0)

R=r

Over M1 (x1, Y1, Z1), M2 (X2,y2, Z2),

M3 (x3, Y3, Z3) three-point round equation:

2. Tangent equation of Circle

The tangent equation of the 1º circular x²+y²=r² on a point m (x0, y0) is

X0x+y0y=r2

The tangent equation of the 2º circular x²+y²+2mx+2ny+q=0 on a point m (x0, y0) is

X0x+y0y+m (x+x0) +n (y+y0) +q=0

3. The intersection of two circles and the orthogonal conditions

Round c1:x²+y²+2m1x+2n1y+q1=0

Round c2:x²+y²+2m2x+2n2y+q2=0

The intersection θ of two circular c1,c2 is the angle at which the two tangent lines at the intersection point

By the upper formula available, the circular C1 and the C2 are orthogonal to the conditions:

2m1m2+2n1n2-q1-q2=0

§2 Ellipse

1. Basic parameters of Ellipse

As shown in the figure,

Axes (axis of symmetry): Long axis ab=2a

Short shaft cd=2b (a>b>0)

Vertex: A,b,c,d

Center: G

Focus: F1,F2

Focal length: f1f2=2c,

Centrifugal Rate: e=c/a<1

Compression factor: Μ=b/a,μ2=1-e2

Focus Parameters: p=b2/a (the length of the string that is over the focus and perpendicular to the long axis, i.e. the length of the f1h in the graph)

Focus Radius: R1,r2 (a point m (x,y) from the ellipse to the distance between two focuses, i.e. the length of the mf1,mf2 in the graph)

Diameter: PQ (through the chord of the center of the ellipse)

Conjugate diameter: diameter slope is k,k ', and meet KK ' =-B2/A2

Alignment: Straight line L1 and L2 (parallel to short axis, to short axis distance to a/e)

2. The nature of the ellipse

The sum of the ⑴ radius is constant (equal to the long axis 2a): R1+R2=2A. The ellipse is the locus of the moving point m of the constant (the long axis) of the distance between the two fixed-point (i.e. the focus).

⑵ the trajectory of the moving point m of a constant (centrifugal rate e) that is less than 1 of the distance between a certain point (one of the focal points) and a distance to a certain line (a guideline), as shown above, Mf1/me1=mf2/me2=e or written as r1/me1=r2/me2=e.

The ⑶ ellipse is obtained by compressing the circle of radius A to a proportional μ=b/a (compression factor) along the Y axis, where point M (x,y) on the circular x2+y2=a2, then the point M ' (X,y ') in the ellipse

On, where Y ' =μy.

Any diameter of the ⑷ ellipse is divided equally between the strings parallel to its conjugate diameter, if the length of the two conjugate diameters is 2a1 and 2b1 respectively, and the angle of the long axis (acute angle) is α and β, then there are

The ⑸ of the focus radius (R1R2) of a point M (x,y) of an ellipse is equal to the square of its corresponding half conjugate diameter (no in the figure).

⑹ set mm ', NN ' for the ellipse of the two conjugate diameters (such as above), through the M,m ' respectively for a straight line parallel to NN ', and through the N,n ' respectively for a straight line parallel to the MM ', then the four lines constitute a parallelogram area of a constant 4ab.

3. Elliptic equation

Legend

Equation

Vertex, center, focus, long half axis, short half axis

① Standard equation

(a>b>0)

② parameter equation

Vertex:

A,b (±a,0)

C,d (0,±B)

Center: G (0,0)

Focus: F1,f2 (±c,0)

Long Half axis: a

Short half axis: b

①

(a>b>0)

② parameter equation

Vertex:

A,b (G±A,H)

C,d (G,H±B)

Center: G (G,H)

Focus:

F1,F2 (G±C,H)

Long Half axis: a

Short half axis: b

(a>b>0)

Vertex:

A,b (0,±a)

C,d (±b,0)

Center: G (0,0)

Focus:

F1,F2 (0,±c),

Long half axis: A; short half axis: b

Polar coordinate equation:

(e<1)

The pole is a focus (F1), and the polar axis points from the focus to the nearest vertex.

Long Half axis:

Short Half axis:

Focal length:

4. Tangent of an ellipse

Set Ellipse:

The tangent Mt equation of a point m (x0,y0) in the ellipse is

If the slope of the tangent MT is k, then the tangent equation is

The positive sign in the type represents the two tangents of the diameter mm ' ends point (m and M ').

The Tangent MT Outer (∠F1MH) between the two focal radii (MF1,MF2) of point M, which is α=β in the graph, and

And the normal nm of point m splits the internal angle (∠F1MF2) equally.

§3 Double curve

1. Basic parameters of Hyperbola

Vertex: a,b

Center: G

Focal length:

(The length of the chord that is over the focus and perpendicular to the axes, i.e. the length of the f1h in the graph)

(The distance of a point m (x,y) from a hyperbolic curve to two focal points, as in the figure MF1,MF2)

Diameter: PQ (through the chord of the hyperbola Center)

Conjugate diameter: = diameter slope is k,k ' respectively, and satisfies

Alignment: Straight line L1 and L2 (perpendicular to axes, distance to center is a/e)

Asymptote

2. The properties of Hyperbola

(1) The difference of the focus radius is constant (equal to axes 2a):

That is, a hyperbola is the locus of the moving point m of a constant (axes) distance to the two point (i.e. the focus) (so that the r1-r2=2a points are in one of the hyperbola, and the r2-r1=2a points belong to the other of the hyperbolic curves).

(2) as shown above,

That is, a hyperbola is the locus of the moving point m of a constant (centrifugal rate e) that is greater than 1 in the distance from a certain point (one of the focal points) to a definite line (a guideline).

(3) any diameter of the hyperbola halves the string parallel to the conjugate diameter. If the length of the two conjugate diameters is 2a1,2b1, the two diameters and the axes angles (acute angles) are α and β (α<β) respectively, then

(4) The product of the focal radius of one point m of a hyperbola is equal to the square of its corresponding half conjugate diameter.

(5) Set mm¹,nn¹ as two conjugate diameters of hyperbola, which are parallel and nn¹ by M,m¹ respectively, and through N, M¹ respectively as straight line parallel to mm¹, then the area of these four lines constitutes a constant 4ab.

3. Hyperbola equation

Graphics

Equation

Vertex, center,

Focus, Asymptotic line

1, Standard equation

2, parametric equation

Vertex:

Center: G (0,0)

Focus:

Asymptote

(with

Conjugate hyperbola)

Center: G (0,0)

Focus:

Vertex:

Center: G (G,H)

Focus:

Asymptote

Polar coordinate equation:

(The pole is at a focal point, and the polar axis is a ray from the focus back to the vertex, and this equation gets one of the two curves, the other can be obtained by hyperbola)

Axes

Virtual axis:

Focal length:

Vertex a,b:

Center: G (0,0)

Focus

Shaft Length:

Asymptote: x=0,y=0

Vertex a,b:

Center:

Shaft Length:

Asymptote

4. Tangent of Hyperbola

(1)

，

Tangent of a point m (x0,y0) on a hyperbola (MT)

Equation is:

If the slope of the tangent MT is k, then the tangent equation is:

The positive sign in the type is in the diameter mm¹ ends point

Two tangents of (M and m¹).

Tangent MT put M dot two focal radius between the inner angle

(∠F1MF2) split evenly, in figure

And the normal of the M point Outer (∠F1MH) equally.

(2)

Tangent segments between two asymptotic lines TT1 tangent m

(TM=MT1), and the area of the OTT1 (shadow portion of the figure)

Area of a parallelogram ojmi (shaded part of the figure)

From:http://202.113.29.3/nankaisource/mathhands/