Analytic geometry: Fifth chapter two times curve (2) parabola General two times curve _ analytic geometry

§4 parabola

1. Parabola Basic parameter

As shown in the figure:

Axis: AB

Vertex: A

Focus: F

Focus parameter: p (the length of the string that is over the focus and perpendicular to the axis, that is, half of the CD in the picture)

Focus Radius: R (the distance from point to focus on the parabola, as in the figure MF)

Diameter: Straight line EMH (parallel to Parabola line)

Alignment: Line L (perpendicular to axis, distance to vertex A is P/2, distance to focus F is P)

2. Properties of Parabolic line

As pictured above, Mf=me, or written as R=me. That is, the trajectory of a moving point m that is equal to the distance from a certain point (focus) to a certain line (alignment).

3. Parabolic equation

Graphics

Equation

Vertex, Focus, alignment

1 Standard equation

2 Polar coordinate equation

(pole is in focus F, polar axis coincides with parabola axis, back vertex)

Vertex: A (0,0)

Vertex: A (0,0)

Vertex: A (0,0)

Vertex: A (0,0)

Vertex: A (g,h)

Vertex: A (g,h)

Vertex:

(When a>0, openings

Up

(When a<0, openings

Down

Focus Parameters:

The intersection of the X axis a1,a2:

1.

(a>0)

2. Parametric equation

(a>0)

Vertex

Focus Parameters:

4. Tangent line of parabola

(1)

The tangent (MT) equation is:

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

The angle between the focus radius of the tangent MT and the diameter of the M point

And everything that is parallel to MT is halved by the diameter of the M-point.

(2)

The angle between the two tangent lines of a parabola is equal to half the angle of the focus radius of the two tangent,

(3) from the focus F as a parabola at the tangent line of the point M, then the pedal trajectory is the tangent of the vertex.

§5 General two times curve

1. Equation of general two times curve

The two-time equation of x,y

The curve represented is called a general two-time curve.

2. The general properties of two times curve

(1) Intersection point of Line and two times curve

A line with a two-time curve at two o'clock (real, imaginary, coincident)

(2) two times the diameter of the curve and the center

The midpoint of a chord in a two-time curve parallel to the known direction is on the line, which is called the diameter of the two-time curve, he shares a set of strings, and the number of directions in the given direction is α,β, the equation of the diameter is

or rewritten as

Thus, the diameter of the two-time curve consists of a straight beam, Shanne any diameter through the following two straight points:

At this time, all the diameters of the two curves pass through the same point, called the center, which is called the Heart two times curve,

The coordinates of the center are

<1>

Then the curve has no center;

<2>

At this point the curve has an infinite center, that is, the center is on the same line (the center line), the two curves are called the unintentional two-time curve.

(3) two times the spindle (or axis of symmetry) of the curve

If the diameter is perpendicular to the string that is divided by it, it is called the spindle (or symmetry axis) of the two-times curve, and the centerless two-time curve has a solid spindle, and the two-time curve has two solid axes, they are perpendicular to each other, and the focus is the center.

3. Tangent lines and normals on two-time curves

The tangent equation of one point m (xo,yo) on the two quadratic curve is

The line perpendicular to the tangent of the point M to the two-time curve is called the normal at point M, and its equation is

4. Invariant of two times curve

By the equation of general two times curve

(1)

The following three functions are composed of the coefficients:

The invariant of a two-time curve, that is, after a coordinate transformation, these quantities are invariant.

Determinant D is called a discriminant of the two-second equation (1).

5. Standard equation and shape of two times curve

Not variable

The standard equation under the coordinate transformation

Curve shape

Yes

Heart

Two

Times

Song

Line

In-style

A,c is a characteristic equation

The two characteristic roots

Ellipse when ds<0

Virtual Ellipse when ds>0

There is a public real point of

A pair of dashed lines

Double curve

Intersect two straight lines

No

Heart

Two

Times

Song

Line

In-style

Parabolic

is parallel to the two straight lines,

Is coincident with two straight lines,

A pair of dashed lines when

6.

A

Graphics

Vertex • Center • Focus parameters

Parabolic

Elliptic

Vertex:

Double curve

7. Cone Cut Line

The two-time curve is the cutting line of the normal conical plane, so the two-time surface is also called the cone cut line.

When using a plane p to cut a positive cone, if p does not pass the cone top, and does not parallel with any bus, then the truncated line is elliptical;

If P does not pass the cone top, while parallel with a bus, the cut-off line is a parabolic line, if p does not pass the cone top, and parallel to two bar bus, the cut line is a hyperbola, if p is perpendicular to the cone axis, the cut-off line is round.

If p passes through the cone top, then the ellipse becomes a little, the hyperbola becomes a pair of intersecting lines, and the parabola becomes a straight line of p and cone tangent.

from:http://202.113.29.3/nankaisource/mathhands/