Methods of solving problems in analytic geometry in senior high school mathematics Let's first analyze the trend of the proposition of analytic geometry in the college entrance examination:
(1) The question type is stable: In recent years, the analytic geometry questions in the college entrance examination have been stable at 3 (or 2) multiple-choice questions, 1 fill-in-the-blank questions and 1 answer questions, accounting for about 20% of the total score.
(2) Overall balance, with outstanding emphasis: there is almost no omission in the examination of the knowledge of straight lines, circles and conic curves. Through the reorganization of knowledge, we should not only pay attention to comprehensiveness, but also highlight key points, so as to ensure that the main knowledge supporting the mathematical knowledge system has a high proportion and maintain the necessary depth. In recent years, the examination of analytic geometry in the college entrance examination of new textbooks mainly focuses on the following categories:
① Find the curve equation (the type has been determined and the type is to be determined);
(2) Intersection point of straight line and conic curve (including tangent);
③ Maximum (extreme value) problem related to curve;
(4) Geometric verification related to curves (symmetry or finding symmetric curves, parallel and vertical);
⑤ Explore the numerical characteristics between geometric quantities and parameters in the curve equation;
(3) The conception and infiltration ability of mathematical thought: Although some basic questions are very common, if we use the idea of combining numbers and shapes, we can get the answers quickly and correctly.
(4) The question type is novel and the position is uncertain: in recent years, it is difficult to analyze geometry questions, multiple-choice questions and fill-in-the-blank questions are easy to average, and the solution questions may not be in the finale position, so the amount of calculation is reduced and the amount of thinking is increased. Strengthen the connection with related knowledge (such as vectors, functions, equations, inequalities, etc. ), and highlight the ability requirements of inquiry learning in the teaching materials. Increase the weight of exploratory questions.
In the recent college entrance examination, the examination of straight lines and circles is mainly divided into two parts:
(1) Use multiple-choice questions to examine the basic concepts and properties of this chapter. This kind of question is generally not difficult, but it must be tested every year. The inspection contents mainly include the following categories:
① Topics related to the concepts in this chapter (dip angle, slope, included angle, interval, parallel and vertical, linear programming, etc.). );
(2) the solution of memory blindness (including point symmetry and straight line symmetry);
③ For the problems related to the position of a circle, the conventional method is to study the interval from the center of the circle to a straight line.
And other "standard parts" types.
(2) It is comprehensive and difficult to investigate the positional relationship between straight lines and conic curves by solving problems.
It is expected that in the next year or two, the exams in this chapter of the College Entrance Examination will remain relatively stable, that is, there will be no major changes in the types, quantity, difficulty and key exam contents.
Comparatively speaking, the content of conic is the core content of plane analytic geometry, so it is the key content of college entrance examination. There are generally 2 ~ 3 objective questions and 1 analytic questions in the annual college entrance examination papers, which are easy, medium and difficult. The main contents include the concept and properties of conic curve, the positional relationship between straight line and cone, etc. Judging from the college entrance examination questions in the past ten years, there are roughly the following three categories:
(1) Study the concept and properties of conic curve;
(2) Find the curve equation and trajectory;
(3) The problem about the position relation of straight line, circle and conic curve.
Multiple-choice questions mainly focus on ellipses and hyperbolas, fill-in-the-blank questions focus on parabolas, and solutions focus on the positional relationship between straight lines and conic curves. For solving curve equations and trajectories, the college entrance examination generally does not give graphs to test students' imagination and ability to analyze problems, thus embodying the basic ideas and methods of analytic geometry. Generally, the circle is not examined separately, but is always a comprehensive problem combining straight lines and conic curves. Equilateral hyperbola is basically not a problem, and the translation or translation of coordinate axes generally simplifies the equation, which mostly appears in the form of multiple-choice questions. The solution of analytic geometry is generally difficult. In recent two years, we have investigated the basic methods of analytic geometry-coordinate method and the propositional trend of the application of conic properties, in order to attract our attention.
Please pay attention to the application of the definition of conic curve in solving problems and some properties of the background plane geometry studied by analytic geometry. Judging from the examination questions in the past two years, analytic geometry has a tendency to move forward, which requires candidates to work harder on basic concepts, methods and skills. Parametric equation is an auxiliary tool to study curves. The college entrance examination questions involve more mathematical thinking methods of mutual transformation and equivalent transformation between parameter equations and constant equations.
The focus of investigation should be on the trajectory equation and the positional relationship between straight line and conic curve, and the equivalence relationship is often established by simultaneous and elimination of the equations of straight line and conic curve, with the help of the generation of Vieta theorem and vector bridge. The knowledge topics involved in the examination questions include finding the curve equation, the range of parameters, the maximum value, the fixed value, the straight line passing through the fixed point, blindness, etc., so we should master the basic solutions of these topics.
The proposition pays special attention to the examination of the rigor of thinking, and the following topics need to be considered when solving problems:
1, which coordinate axis to focus on when setting the curve equation; Pay attention to the undetermined form of the equation and the use of the parametric equation.
2. The slope of the straight line exists or does not exist, and the slope is zero. Pay attention to the influence of "D" on the intersection topic.
3. The way to give the conclusion of the proposition: find out whether the small questions given in the topic are parallel questions or progressive questions. If the pre-and post-sub-problems have their own strengthening conditions, it is a parallel relationship, and the sub-problems in front of the conclusion cannot be used; However, the examination questions often give a progressive relationship, including (1), the first question is to find the curve equation, the second question is to discuss the positional relationship between the straight line and the conic curve, (2) the first question is to find the eccentricity, the second question is to find the curve equation by combining the properties of the conic curve, and (3) the exploratory question. When solving problems, we should consider applying different problem-solving skills according to different situations.
4. If the topic conditions are combined with vector knowledge, we should also pay attention to the given form of the vector:
(1), which directly reflects the position relationship and properties of graphics, such as? =0, = (), λ, and vector expressions through the "four centers" of a triangle, etc.
(2) = λ: If the coordinates of m are known, use vector expansion; If the coordinates of m are unknown, the coordinates of m point are expressed by the formula of fixed fraction point.
(3) If the topic condition is given by multiple vector expressions, consider its graphic characteristics (combination of numbers and shapes).
5. Consider the difference between the first definition and the second definition of the conic, and pay attention to the application of the properties of the conic.
6. Pay attention to the combination of numbers and shapes, and pay special attention to the plane geometric properties reflected by graphics.
7. Another key point of analytic geometry problems is students' basic computing ability, so students generally find analytic geometry problems difficult to deal with. Therefore, it is necessary for us to discover and accumulate some common formula deformation skills in the usual problem-solving process, such as the separation skills of false fractions, the replacement skills of idiots, the replacement skills of Vieta's theorem to construct symmetric formulas, and the deformation skills of constructing mean inequality, so as to improve the problem-solving speed.
8. Plane analytic geometry and plane vector are combined because of the combination of numbers and shapes. Putting it forward at the intersection of their knowledge points is also a highlight of the college entrance examination proposition. The topic of the positional relationship between straight lines and conic curves is a new and long-lasting examination focus. In addition, the range, maximum, fixed value, blindness and other comprehensive questions of parameters in conic curve are also common questions in college entrance examination. Generally speaking, analytic geometry requires a large amount of calculation, certain skills and careful calculation. In recent years, the difficulty of analyzing geometry has been reduced, but it is still a comprehensive topic, which tests the will quality and mathematical wit of candidates. It is a subject of great disagreement in the college entrance examination.
Example 1 known points A (- 1, 0), B( 1,-1), parabola. O is the origin of coordinates, and the moving straight line L passing through point A intersects parabola C at points M and P, and the straight line MB intersects parabola C at another point Q, as shown in the figure.
(1) If the area of △POM is, find the angle between the vector and.
(2) Try to confirm that the straight line PQ passes through a fixed point.
Although the college entrance examination proposition is ever-changing, as long as we find some corresponding laws, we will boldly guess some ideas and trends of the college entrance examination problem-solving proposition to guide our later review. We should adopt a correct attitude towards the college entrance examination, at the same time, we should pay more attention to the further consolidation of basic knowledge, do more simple comprehensive exercises and improve our problem-solving ability.
First, the college entrance examination review suggestions:
The content of this chapter is the key content of the college entrance examination. The winter equinox of the left glaze, which accounts for 15% of the total score in the annual college entrance examination paper, has remained stable, and generally there are 2-3 objective questions and a solution. Multiple-choice questions and fill-in-the-blank questions not only attach importance to basic knowledge and methods, but also have certain flexibility and comprehensiveness, and most of them are intermediate questions. The answer focuses on the examinee's understanding, grasping and flexible application of basic methods and mathematical ideas, which is comprehensive and difficult. It is often used as a key problem or a finale problem, focusing on the positional relationship between a straight line and a conic curve, and finding the curve equation and the maximum value of the conic curve. Examine the ability of combining numbers and shapes, equivalent transformation, classification discussion, functional equation, logical reasoning and so on. , need higher thinking ability and thinking method.
In recent years, the hot topics of analytic geometry examination are as follows.
―― Find the curve equation or the locus of a point.
-Find out the range of parameters.
-Evaluation domain or maximum value
-positional relationship between straight line and conic curve
The above topics often overlap. For example, the range of parameters should be considered in solving trajectory equation, and the topic of parameter range or maximum value should be combined with the relationship between straight line and conic curve.
Summarize the college entrance examination questions in recent years, and pay attention to the following topics when reviewing:
1, focusing on the definition or properties of ellipse, hyperbola and parabola.
This is because the definitions and properties of ellipse, hyperbola and parabola are the cornerstones of this chapter, and all college entrance examination questions should involve these contents. We should be good at consolidating and strengthening the three basics from multiple angles and levels, and strive to promote the deepening and sublimation of knowledge.
2. Pay attention to find the equation of the curve or the trajectory of the curve.
The equation or trajectory of curve is often the proposition object of college entrance examination, which is difficult. Therefore, we should master the general methods of finding the equation or trajectory of a curve: definition method, direct method, undetermined coefficient method, progressive method (intermediate variable method), correlation point method and so on. And pay attention to the combination of knowledge and interest with vectors and triangles.
3. Strengthen the review of the position relationship between straight line and conic curve.
Because the positional relationship between straight line and conic curve has always been a hot topic in college entrance examination, such topics often involve the properties of conic curve, the basic knowledge points of straight line, the midpoint of line segment, chord length, vertical line and other topics. Therefore, when analyzing the topic, we use the idea of combining numbers and shapes, rather than seeking the connection between chord length formula and Vieta's theorem to solve the problem, which strengthens the examination of various mathematical abilities, especially pays attention to "operation" and enhances the ability of abstract operation and deformation. The idea of solving analytic geometry is easy to analyze, and it is often abandoned because the operation is less than half. In the learning process, through solving problems, seeking fair operation schemes and simplifying basic ways and methods of operation, we can experience the whole process of the occurrence and overcoming of operational difficulties and enhance our confidence in solving complex problems.
4. Pay attention to the feedback and refinement of mathematical thinking methods, so as to optimize the problem-solving ideas and simplify the problem-solving process.
Make good use of equation thought. Most problems in analytic geometry are given straight lines and conic curves in the form of equations, so the chord length of the intersection of straight lines and conic curves can be treated as a whole by Vieta theorem, which can simplify the calculation of solving problems.
Make good use of function thought and master coordinate method.
Second, knowledge combing
● Curve equation or point trajectory.
Finding the trajectory equation of curve is one of the basic topics of analytic geometry, and it is also a hot topic and focus of college entrance examination, which frequently appears in college entrance examinations over the years. Especially in today's college entrance examination reform, students' innovative consciousness is taken as a breakthrough, and students' logical thinking ability, calculation ability, problem analysis and problem solving ability are mainly examined. The popularization of trajectory equation well reflects students' mastery of these abilities.
Here are some common methods.
(1) direct method: the geometric condition satisfied by the moving point itself is the equivalence relation of some geometric quantities. We only need to "translate" this relationship into an equation containing X and which brand of liquid foundation is better than Y, and then we can get the curve trajectory equation.
(2) Definition method: If the trajectory of the moving point conforms to the definition of a basic trajectory, the trajectory equation of the moving point can be directly obtained according to the definition.
(3) Geometric method: If the obtained trajectory satisfies some geometric properties (such as the properties of vertical lines in line segments, bisectors of angles, etc.). ), you can use geometric methods to list geometric formulas, and then the coordinates of points are simpler.
(4) Correlation point method (substitution method): In some topics, it is not convenient to list the conditions that a moving point meets with an equation, but this moving point moves with another moving point (called correlation point). If the conditions satisfied by the relevant point are obvious, then we can use the coordinates of the moving point to represent the coordinates of the relevant point, and then substitute the relevant point into the equation it satisfies, and then we can get the trajectory equation of the moving point.
(5) Parametric method: It is sometimes difficult to obtain the geometric conditions that a moving point should meet, and there are no obvious related points, but it is easy to find that the movement of this moving point is often restricted by another variable (angle, slope, ratio and intercept), that is, the x and y in the coordinates of the moving point (x, y) change with the change of another variable, so we can call this variable a parameter to establish the parameters of the trajectory. The general equation of trajectory can be obtained by eliminating parameters. When selecting parameter variables, we should pay special attention to the influence of their value range on the value range of moving point coordinates.
(6) Trajectory intersection method: When finding the trajectory of a moving point, sometimes there will be a trajectory problem that requires the intersection of two moving curves. This kind of problems often get the coordinates of intersection points (including parameters) by solving equations, and then get the trajectory equation by eliminating parameters. This method is usually used with parameter methods.
● Find the topic of parameter range
In analytic geometry, parameters are often used to describe the movement and change of points and curves. For the discussion of the range of parameter variables, we need to use the mutual transformation between change and invariance, and think with functions and variables. Therefore, under the guidance of the idea of function and equation, we should use the known variable range and the condition of equation root to find the parameter range.
Example 1. Known ellipse C: Try to determine the range of m, so that there are two different points on the ellipse that are symmetrical about the straight line L: y = 4x+m.
Example 2: It is known that the center of the hyperbola is at the origin, the right vertex is A (1, 0), points P and Q are on the right branch of the hyperbola, and the interval from point M (m m, 0) to straight line AP is 1.
(1) If the slope of the straight line AP is k and the range of the real number m is true.
(2) When the heart of Δ δAPQ happens to be point m, find the equation of this hyperbola.
Low-range and most valuable topics
The range, chord length, maximum and minimum values of functions related to analytic geometry are comprehensive problems of analytic geometry and functions, which need to be handled by functions.
In analytic geometry, the relationship between objective and function is generally listed according to conditions, and then parameter method, collocation method and discriminant method are selected according to the characteristics of function relationship, and the maximum or minimum value is obtained by applying the nature of inequality and the maximum value method of trigonometric function. In addition, with the help of graphics, the maximum value can be obtained by number knot.
Example 1, as shown in the figure, it is known that the vertex of the parabola y2 = 4x is O, the coordinate of point A is (5,0), the straight line L with the inclination of π/4 intersects with the line segment OA (but it is point O or point A), and the parabola intersects with two points M and N, so as to find the equation of the straight line when the area of △AMN is the largest.
● The relationship between straight line and conic curve
1. The topic of the position relationship between straight lines and conic curves is transformed from algebraic perspective into the study of the number of real solutions of a system of equations (if numbers and shapes can be combined, it is easier to rely on the geometric properties of graphs). That is to say, when judging the positional relationship between a straight line and a quadratic curve C, we can bring the straight line equation into the equation of curve C and eliminate it to Y (sometimes it is more convenient to eliminate it to X) to get the unary equation ax2+bx+c = 0.
When a=0, this is a linear equation. If the equation has a solution, L and C intersect, and there is only one common point. If c is a hyperbola, then l is parallel to the asymptote of hyperbola; If c is a parabola, then l is parallel to the axis of symmetry of the parabola. So when a straight line and a hyperbola or parabola have only one common point, the straight line and the hyperbola or parabola may intersect or be tangent.
When a≠0, if δ >; 0 l intersects c
δ = 0 l tangent to c
δ& lt； 0 l separated from c
2. The chord length of conic curve is generally solved by chord length formula combined with Vieta theorem.
There are two common methods to solve the midpoint of a chord: one is to use Vieta's theorem and the midpoint coordinate formula; Secondly, the relationship between midpoint coordinates and slope is constructed by using the endpoint on the curve and the coordinates satisfying the equation (point difference method)
The midpoint chord problem is to get a topic to further study the midpoint of the chord when a straight line intersects a conic curve in winter. The midpoint chord problem is an important and hot topic in analytic geometry, which often appears in college entrance examination questions. The "point difference method" is an effective method to solve the mid-point chord problem of conical curve, and as the name implies, it is the method of difference between points. The steps can be briefly described as follows: ① Set the coordinates of the two ends of the chord; (2) substituting endpoint coordinates into a quadratic equation for subtraction; (3) obtain the relationship between the coordinates of the chord midpoint and the slope of the straight line, and then obtain the equation of the straight line; (4) Jane
This paper tries to discuss the solution of a college entrance examination question and talk about some personal opinions.
First, the college entrance examination questions
Ellipse c:+=1(a > b >; The two focal points of 0) are F 1, F2, point p is on ellipse C, and PF 1⊥F 1F2, |PF 1|=, |PF2| =.
(1) Find the equation of ellipse c;
(2) If the center of the straight line L passes through the circle M x2+y2+4x-2y = 0, passes through the ellipse C at points A and B, and B is symmetrical about point M, then the equation of the straight line L is found.
Second, the way to solve the problem
The solution of (1) is not repeated, and the answer is += 1. On this basis, the solution of problem (2) is studied.
1. Using the concept of equation
Let A(x 1, y 1) and B(x2, y2) be known, and the equation of the circle is (x+2)2+(y- 1)2 = 5, then the coordinate of the center m is (-2, 1).
∴y= k(x+ 2)+ 1，+= 1。
(4+9 k2)x2+(36 k2+ 18k)x+36 k2+36k-27 = 0。
∫A, b is symmetric about point m,
∴ =-= -2, the solution is k =.
The equation of the straight line L is: 8x-9y+25 = 0.
2. Use the idea of "point difference method"
It is known that the equation of a circle is (x+ 2)2+ (y- 1)2= 5, so the coordinate of the center m is (-2, 1).
Let A(x 1, y 1) and B(x2, y2) be defined by the meaning of the sum of the questions x 1≠x2.
+ = 1( 1)+= 1(2)
From (1)- (2)
+ = 0(3)
Because A and B are symmetrical about point M, x 1+x2 = -4, y 1+y2 = 2, and k 1 = = is substituted into (3). So the equation of the straight line L is: 8x-9y+25 = 0. After investigation, the linear equation obtained accords with the meaning of the question.
Three, familiar with the two concepts
The operation of train of thought 1 is complicated, especially the step of eliminating elements to get equations, which many students can't pass smoothly. The second idea is simple and easy for students to master. For these two ideas, it is necessary to analyze that the straight line L passes through the center of the circle, and the center of the circle is the midpoint of the chord. These methods are often involved in examination questions.
Fourthly, thinking about the "point difference method"
1. Reflections on the application conditions of "point difference method"
The "slip method" is relatively simple to use, so what are the conditions for using the "slip method"?
Suppose a straight line intersects the curve mx2+ny2 = 1(n, m is a non-zero constant and not negative at the same time) at points A and B. Let A(x 1, x2) and B(x2, y2), then mx12+ny1. It can be seen that if one of them is known, the other one can be found, that is to say, to use the "point difference method", you need to know the midpoint of AB and the slope of AB to find the other one, and then do a simple test.
2. Let me introduce an ingenious and novel method to solve the midpoint chord problem.
Example Given a hyperbola x2-= 1, ask whether there is a straight line L, so that M( 1, 1) is the midpoint of the chord AB of the straight line L cut by the hyperbola. If yes, find the equation of the straight line L; If it does not exist, please explain why.
M( 1, 1) is the midpoint of obvious reading b, which can be set as A( 1+ s, 1+ t), B( 1- s, 1- t), (s,.
( 1+s)2-= 1( 1)( 1-s)2-= 1(2)
(1)+ (2) can get s2= t2 (3)
( 1)- (2) t = 2s (4)
Substituting (4) into (3) can get s= 0 and t= 0, which is impossible, so there is no such straight line.
Here we go back to solving the problem:
It is known that the straight line L intersects the conic curve: ax2+by2 = 1(a, B makes the equation conic) At two points A and B, set the midpoint as M(m, n) to find the straight line L equation.
The thinking of solving problems is set as A(m+ s, n+ t), B(m-s, n-t), (s, t∈T). Since a, b and m do not coincide, it is known that s and t are not all zero. Points A and B are also on hyperbola ax2+by2 = 1. A(m-s)2-b(n- t)2= 1。 Solution: ams = bnt, am2 +s2 = bn2+t2. (Because all the operations here are letters, the expressions are complicated, so I don't want to find out all the concrete forms of the expressions, just talk about ideas.) Further solve the values of S and T, so as to know the values of A and T.