Urgent! What are the laws and theorems of parabola, hyperbola, ellipse and circle in college entrance examination mathematics, and what are the ideas for doing the questions?

First, the ellipse:

(1) Definition of ellipse: the locus of a point whose sum of the distances from a point to two fixed points on a plane is equal to a constant (greater than).

The second definition: the trajectory of a point on a plane, the ratio of the distance from the point to a fixed point to the distance to a fixed straight line is constant.

Among them, two fixed points are called the focal points of an ellipse, and the distance between the focal points is called the focal length; A straight line is called a directrix.

This constant is called eccentricity.

Note: indicates ellipse; Represents a line segment; No track;

(2) Standard equation, image and geometric properties of ellipse:

The center is at the origin and the focus is on the axis.

Standard equation

The parametric equation is a parameter)

As a parameter)

graphic form

pinnacle

Axis of symmetry; The minor axis is, and the major axis is.

focus

focal distance

Eccentricity (the greater the eccentricity, the flatter the ellipse)

Collinear straight line

Diameter (focal length)

Focus radius

Focus chord

It is only related to its midpoint abscissa.

It is only related to the ordinate of its midpoint.

focal distance

Second, hyperbola:

(1) Definition of hyperbola: the trajectory of a point, and the absolute value of the difference between the distance of the point and two fixed points on the plane is equal to a constant (less than).

The second definition: the trajectory of a point on a plane, the ratio of the distance from the point to a fixed point to the distance to a fixed straight line is constant.

Among them, two fixed points are called focal points of hyperbola, and the distance between focal points is called focal length; A straight line is called a directrix.

This constant is called eccentricity.

Note: and () represent a hyperbola.

Represents two rays; No track;

(2) The standard equation, image and geometric properties of hyperbola:

The center is at the origin and the focus is on the axis.

Standard equation

graphic form

pinnacle

Axis of symmetry; The imaginary axis is, and the real axis is.

focus

focal distance

Centrifugal rate (the greater the centrifugal rate, the larger the opening)

Collinear straight line

Asymptote

Diameter (focal length)

The focal radius is in the left branch.

The branch on the right

Branch at the next level

Superior branch

focal distance

(3) Asymptote of hyperbola:

① Finding the asymptote of hyperbola can make 1 on the right be 0, that is, factorization.

(2) The equation of hyperbolic system with hyperbolic asymptote is:

(4) The equilateral hyperbola is, and its eccentricity is.

Third, parabola:

(1) Definition of parabola: The distance from a fixed point on a plane to that point is equal to the trajectory from the distance point to a fixed straight line.

Among them, the fixed point is the focus of parabola, and the fixed line is called directrix.

(2) Standard equation, image and geometric properties of parabola:

The focus is on the axis,

The focus of the right opening is on the axis,

Open to the left, focus on the axis,

The upward focus of that open is on the axis,

Open down

Standard equation

graphic form

pinnacle

axis of symmetry

axis