Mastering and flexibly using the midpoint coordinate formula is the key to deal with this kind of problem.
The symmetry problem of a point about a straight line is a generalization of the symmetry problem of a point about a point. To deal with this kind of problem, we mainly grasp two aspects: ① The product of the slope of the two-point connecting line and the known straight line is equal to-1; ② The midpoint of two points is on a known straight line.
The symmetry problem of a straight line about a point can be transformed into the symmetry problem of a point on a straight line. It should be noted that two symmetrical straight lines are parallel.
We often use parallel straight line system to solve it.
example
Find the equation (0, 1) of the line 2x+1y+16 = 0 which is symmetric about point p.
analyse
This problem can be solved by using two parallel lines, and the distance from point P to two straight lines is equal. You can also take a point on a known straight line first, and then find the symmetrical point of the point about point P, and substitute it into the symmetric straight line equation to determine the correlation constant.
Solution 1
According to the central symmetry property, the symmetrical straight line is parallel to the known straight line, so the equation of the symmetrical straight line can be set as 2x+ 1 1y+c=0.
The distance formula from point to straight line is obtained.
That is, | 1 1+c|=27, and c= 16 (i.e. known straight line discarding) or c=
-38.
So the equation of symmetrical straight line is 2x+ 1 1y-38=0.
Solution 2
Take two points A (-8,0) on the straight line 2x+1 y+16 = 0, then point A (-8,0) is about the symmetrical point B (8 8,2) of p (0 0,01).
According to the central symmetry property, the symmetrical straight line is parallel to the known straight line, so the equation of the symmetrical straight line can be set as 2x+ 1 1y+c=0.
Substitute b (8 8,2) to get c=-38.
So the equation of symmetrical straight line is 2x+ 1 1y-38=0.
comment
The first solution is to use the fact that a symmetrical straight line must be parallel to a known straight line, and then the distance between the point (symmetry center) and the two straight lines is equal. The second solution is to transform it into a point-to-point symmetry problem, get the coordinates of the symmetrical point by using the midpoint coordinate formula, and then write the linear equation by using the linear equations.
Both solutions to this problem reflect the superiority of linear system equations.
There are two cases about the symmetry of straight lines: ① two straight lines are parallel; ② Two straight lines intersect.
For ①, we can solve it by transforming it into a symmetry problem about a straight line; For ②, the general solution is to find the intersection first, then use "diagonal", or turn it into a symmetry problem of a point about a straight line.
example
Find the equation of the symmetrical line L: X-Y+ 1 = 0.
analyse
According to the meaning of the question, the two straight lines l 1 and l2 are parallel straight lines. Solving this symmetry can be transformed into a symmetry problem between a point and a straight line, and then solved by a parallel straight line system or by equal distance.
solve
According to the analysis, let the equation of the straight line L be x-y+c=0, and take the point M (1, 0):X-Y- 1 = 0 on the straight line L/kloc-0, then it is easy to get the symmetrical point N: x-y+/kloc-0 about the straight line L2.
Substitute the coordinates of n into the equation x-y+c=0, and get c=3.
So the equation of the straight line L is x-y+3=0.
comment
Transformation symmetry problem is our basic method to solve this kind of problem.
In addition, this problem can also be written in the form of a straight line L by the equation of parallel straight line system, and then the correlation constant of any point on the straight line l2 can be determined according to the fact that the distance from this point to two symmetrical straight lines is equal.