First, the theoretical overview of the method of combining numbers and shapes to solve problems.
(A) Method explanation
First of all, the definition of analytic geometry refers to a small branch of geometry, which mainly studies the relations and properties between set objects by algebraic method, so it is also called "coordinate geometry". It includes two parts: plane analytic geometry and solid analytic geometry, in which plane analytic geometry is analytic geometry in two-dimensional space; Solid analytic geometry is analytic geometry in three-dimensional space, and solid analytic geometry is more complex and abstract than plane analytic geometry.
Secondly, the interpretation of the combination of numbers and shapes refers to the one-to-one correspondence between numbers and shapes in the conditions given in the title, which combines the quantitative relationship between complex and abstract mathematical languages and conditions with simple and intuitive geometric figures. Through the combination of image thinking and abstract thinking, shape helps figures or numbers solve shapes, thus simplifying complex problems and concretizing abstract problems, thus optimizing the way to solve problems.
(B) ideas to solve the problem
When you encounter analytic geometry, you can clearly understand the relationship between conditions and problems, and you can quickly find the breakthrough point by corresponding "number" to "shape". In fact, when you master the method of combining numbers with shapes and can draw inferences from others, all the problems you encounter are the same. Therefore, in order to master the combination of numbers and shapes, the following relations must be clarified: 1. Concepts such as complex numbers and trigonometric functions based on geometric conditions and geometric elements; Second, the obvious geometric meaning contained in the structure of the equation or algebraic equation given in the title; Third, the correspondence between function and image; The correspondence between the fourth point curve and the equation; Fifth, the correspondence between real numbers and points on the number axis.
Second, analyze the combination of "number" and "shape" to solve geometric problems
(A) the circle problem in analytic geometry
Practice has proved that the combination of numbers and shapes is very helpful to solve the circle problem quickly, because in the general process of solving problems, the analytic geometric circle problem mainly focuses on finding the position relationship between circles, the position relationship between circles and straight lines, and the standard equation of circles. For example, when judging the positional relationship between a circle and a straight line, you can intuitively observe that the straight line is outside the circle by establishing a rectangular coordinate system, but you need to write the exact answer steps to score. At this time, it is necessary to have the guidance of "number" and "shape", combined with the idea of solving problems by number-solving problems by shape: by calculating the distance from the center of the circle to the straight line, the distance is greater than the radius of the circle, indicating that the straight line is outside the circle. This is the most basic method to solve the circular problem by combining "number" with "shape". For a more detailed explanation, the following examples illustrate the problem of quickly solving geometric circles by combining "number" with "shape":
Example 1: It is known that the curve y= 1+√(4-x2) and the straight line y=k(x-2)+4 intersect at two different points, and the range of the number k is realistic.
Analysis: Transform the curve y= 1+√(4-x2) to get x2+(y-1) 2 = 4 (1≤ y ≤ 3), which shows that the curve is centered at point A (0,1) and has a radius of 2.
The straight line y=k(x-2)+4 passes through the fixed point b (2 2,4); When the straight line rotates clockwise around point B until the straight line is tangent to the circle, when an intersection point between the straight line and the circle meets the requirements of the topic, it meets the meaning of the topic;
The intersection point m is on the straight line y= 1, so the coordinates of the point m can be calculated, that is, m (-2,1);
The straight line BM can be calculated by the point inclination method, for example, 1kMB=3/4, that is, the distance between point M and point A is equal to the radius;
The equation ∣ 1+2k-4∣/√( 1+k2) can be solved as kBT=5/ 12. Therefore, k ∈ (5/12,3/4].
(B) Analytic geometric inequality problem
Using the combination of numbers and shapes to solve the inequality problem in analytic geometry is mainly to solve the original inequality, which can usually be solved into a curve equation, and then the curve equation is expressed on the number axis. In the calculation process, pay attention to the range and the definition domain, and then the intersection of several figures is the solution set of inequality.
Third, the conclusion
Based on the above, it can be seen that the rational use of "number" and "shape" methods has considerable advantages in the speed and accuracy of solving analytic geometry problems, which can not only reduce the amount of calculation, but also significantly save the time for answering questions and improve the accuracy of solving problems.