1, the definition of a circle: the set of points whose distance to a point on a plane is equal to a fixed length is called a circle, the fixed point is the center of the circle, and the fixed length is the radius of the circle.

(1) standard equation, center and radius r;

(2) General equation

At that time, the equation represented a circle. At this point, the center is and the radius is.

At that time, I said a point; At that time, the equation did not represent any graph.

(3) Method of solving cyclic equation:

Generally, the undetermined coefficient method is adopted: first determine, then solve. Determining a circle requires three independent conditions. If the standard equation of a circle is used,

Demand a, b, r; If you use general equations, you need to find d, e, F e, f;

In addition, we should pay more attention to the geometric properties of the circle: for example, the vertical line of a chord must pass through the origin, so as to determine the position of the center of the circle.

3. Summary of two compulsory knowledge points in high school mathematics: the positional relationship between straight line and circle:

The positional relationship between a straight line and a circle includes three situations: separation, tangency and intersection:

(1) Set a straight line and a circle, and the distance from the center of the circle to L is, then there is; ;

(2) Tangent to a point outside the circle: K does not exist, so to verify whether K exists, set an oblique equation, and solve K with the distance from the center of the circle to the straight line = radius, and get two solutions of the equation.

(3) The tangent equation of a point passing through a circle: circle (x-a)2+(y-b)2=r2, and a point on the circle is (x0, y0), then the tangent equation passing through that point is (x0-a) (x-a)+(y0-b) (y-b) =

4. The positional relationship between circles: it is determined by comparing the sum (difference) of the radii of two circles with the distance (d) between the center of the circle.

Set a circle,

The positional relationship between two circles is usually determined by comparing the sum (difference) of the radii of the two circles with the distance (d) between the center of the circle.

At that time, the two circles were separated, and there were four common tangents at this time;

At that time, the two circles were circumscribed, and the connection line crossed the tangent point, with two outer tangents and one inner common tangent;

At that time, the two circles intersect, and the connecting line bisects the common chord vertically, and there are two external tangents;

At that time, two circles were inscribed, and the connecting line passed through the tangent point, and there was only one common tangent;

At that time, two circles included; It was concentric circles.

Note: when two points on the circle are known, the center of the circle must be on the vertical line in the middle; It is known that two circles are tangent and two centers are tangent to the tangent point.

5, spatial point, straight line, plane position relationship

Axiom 1: If two points of a straight line are on a plane, then all points of the straight line are on this plane.

Application: judging whether a straight line is in a plane.

Express axiom1in symbolic language;

Axiom 2: If two non-coincident planes have a common point, then they have one and only one common straight line passing through the point.

Symbol: Plane α and β intersect, the intersection line is A, and it is denoted as α ∩ β = A. 。

2. Summary of key knowledge points of senior two mathematics.

A, the main grasp of random events (345)

Three operations of (1) event: union (sum), intersection (product) and difference; Note that the difference A-B can be expressed as the product of the reciprocal of A and B.

(2) Four operating laws: exchange law, correlation law, distribution law and democritus law.

(3) Five relationships of events: inclusion, equality, mutual exclusion (mutual incompatibility), opposition and mutual independence.

Second, the definition of probability

(1) Statistical definition: the frequency is stable near a number, which is called the probability of an event; (2) Classical definition: it is required that there are only a limited number of basic events in the sample space, and the possibility of each basic event is equal, then the ratio of the number of basic events contained in event A to the number of basic events contained in the sample space is called the classical probability of events;

(3) Geometric probability: there are infinite elements in the sample space, and the probability of each element is equal, so the sample space can be regarded as a geometric figure, and event A can be regarded as a subset of this figure, and its probability can be calculated by the ratio of the size of the subset figure to the size of the sample space figure;

(4) Axiomatic definition: any mapping from a subset set of sample space to [0, 1] that satisfies three axioms.

Third, the nature and formula of probability

(1) addition formula: P(A+B)=p(A)+P(B)-P(AB), especially if a and b are incompatible with each other, then p (a+b) = p (a)+p (b);

(2) Difference: P(A-B)=P(A)-P(AB), especially if B is included in A, then P (a-b) = P (a)-P (b);

(3) Multiplication formula: P(AB)=P(A)P(B|A) or P(AB)=P(A|B)P(B), especially if A and B are independent of each other, then P (AB) = P (A) P (B);

(4) Total probability formula: P(B)=∑P(Ai)P(B|Ai). This is the result of the cause,

Bayesian formula: p (aj | b) = p (aj) p (b | aj)/∑ p (ai) p (b | ai).

If event B can occur (cause) A 1, A2, ..., An in various situations, then the probability of B's occurrence is calculated by the full probability formula; If event B has occurred, you need the probability that it is caused by Aj, and then use Bayesian formula.

(5) binomial probability formula: pn (k) = c (n, k) p k (1-p) (n-k), k = 0, 1, 2, ..., n. When a problem can be regarded as an N-fold shell hard test (three conditions

3. Summary of key knowledge points of senior two mathematics.

I. Event 1. Inevitable events under SS conditions.

2. Under condition S, an event that will never happen is called an impossible event relative to condition S. 。

3. Random events under 3.SS conditions.

Second, probability and frequency.

1. Using probability to measure the probability of random events can provide a key basis for our decision-making.

2. Repeat the test for n times under the same condition S, and observe whether the event A appears, which is called the number of times that the event A appears in the test nA for n times.

NA is the frequency of event A, and the ratio fn(A)= the frequency of event A. 。

3. For a given random event A, due to the frequency fn(A)P(A), P(A).

Third, the relationship and operation of events.

Four, several basic properties of probability

1. Probability range:

2. The probability of inevitable events P(E)=3. Probability of impossible events P(F)= 1

4. Probability addition formula:

If event A and event B are mutually exclusive, then P(AB)=P(A)+P(B).

5. Probability of opposing events:

If event A and event B are opposite, then AB is an inevitable event. P(AB)= 1，P(A)= 1-P(B)。

4. Summary of key knowledge points of senior two mathematics.

I. Mapping and Function: (1) The concept of mapping:

(2) One-to-one mapping:

(3) The concept of function:

Second, the three elements of function:

Method for judging the same function:

① Corresponding rules;

(2) Domain (two points must exist at the same time)

Solution of resolution function (1):

(1) Define method (together):

② Alternatives:

③ undetermined coefficient method:

④ Distribution method:

(2) The solution of functional domain:

(1) The universe with parameters should be discussed by classification;

(2) For practical problems, after finding the resolution function; We must find its domain, and the domain at this time should be determined according to the actual meaning.

(3) The solution of function value domain:

① Matching method: transform it into a quadratic function and evaluate it by using the characteristics of the quadratic function; Often converted into:;

(2) Reverse solution: the value range used to represent, and then the value range obtained by solving the inequality; Commonly used to solve, such as:

(4) Substitution method: transforming variables into functions of assignable fields and returning to ideas;

⑤ Triangular Bounded Method: Transform it into a function containing only sine and cosine, and use the boundedness of trigonometric function to find the domain;

⑥ Basic inequality methods: transformation and modeling, such as: using the average inequality formula to find the domain;

⑦ Monotonicity method: The function is monotonous, and the domain can be evaluated according to the monotonicity of the function.

⑧ Number-shape combination: According to the geometric figure of the function, the domain is found by the method of number-shape combination.