I believe that readers are familiar with the conventional spread method, first look at the following method of handling the spread, if a, b are two points on the ellipse, the coordinates are (x1, y1), (x2, y2):
Observing the above spread practice, the two equations are multiplied by (y2) and (y1) respectively, and two items appear on the left side of the equation, which are x1y2+x2y1, which has no specific geometric meaning in high school mathematics, but x1y2-x2y1 has a specific geometric meaning. I said earlier that there is a formula for finding the area of a triangle in the form of a point operation, and the formula is derived from the cosine theorem of the product of vector quantities, and the process will not be deduced here, as follows:
As can be seen from the above figure, we can directly write the intercept between the line and the x-axis according to the coordinates of the two points on the line, and the intercept expression exists in the x1y2-x2y1 term, but it should be noted that the y above and below the fraction must correspond, otherwise it will be reversed, and the intercept of the line on the y-axis can also be directly written according to the two-point coordinates.
Therefore, if there is an intercept with the x-axis or y-axis in the problem, you can get the term x1y2-x2y1, and then according to the above-mentioned spread method, you can also find the term x1y2+x2y1 after substitution, and the combination of the two formulas can find x1y2 and x2y1, and the conversion form of the point coordinate can be obtained.
Based on this, we will learn a method of finding a moving straight line without linking it to a fixed point, and first give the following three questions:
According to the symmetry, it can be judged that the fixed point of ab' constant is on the x-axis, so according to a, b'The coordinates of the two points can write the intercept of the straight line on the x-axis, and it is only necessary to prove that the intercept expression is a fixed value, and the existence of x1y2+x2y1 in the numerator of the intercept expression of the intercept can be expressed by using the spread method and the intercept of the known ab on the x-axis, and the upper and lower parts can be reduced. Compared with the traditional point linkage to find geometric relations, this method is faster in solving the problem, and the problem can be written quickly after using the spread method to represent x1y2+x2y1. 
This problem is similar to the previous question, according to the symmetry, it can be judged that the point where the line MP is constant is on the x-axis, and the intercept of the straight line MP on the x-axis can be expressed according to the coordinates of m and p, and the intercept can be proved to be a fixed value. If there is an item x1y2 in the intercept expression, x1y2 can also be obtained according to the combination of the straight line mn and the spread method of the known intercept, and the intercept can be proved to be a fixed value and the straight line mp is over the fixed point.
The third question is different from the first two questions, it is impossible to judge that the vertex of the straight line l is on the coordinate axis, and there are too many moving points in this question, and the known ab slope needs to be transformed to the coordinates of m and n, and it is difficult to convert from only one known slope. After setting the coordinates, the expression of x1y3-x3y1 can be obtained according to the coordinates and intercept formula of point p, x1y3+x3y1 can be obtained according to the point difference, and the expression of x1y3 and x3y1 can be obtained by x1, y1, and y1 can be used to represent the coordinates of point a, and the coordinates of point b can be expressed by x2 and y2, and then the relationship between the slope and the intercept in the l equation can be obtained according to the known slope of ab, and the constant fixed point can be obtained.
There are two functions of the spread method, one is the overall elimination of the element and the elimination of parameters, which is more widely used in the fixed-ratio spread method, and the other is the relationship between the transformation coordinates, and these two purposes have a large amount of calculation with conventional methods, it is recommended to master this method.