Special topic: new definition.

Analysis: firstly, y=x+m-2 can be obtained according to the meaning of the question, and then the value of m can be obtained according to the analytical formula of the proportional function: y=kx(k≠0), and the value of m can be substituted into the equation about x, and then the fractional equation can be solved.

Answer: solution: according to the meaning of the question: y=x+m-2,

∫ The linear function of correlation number [1, m-2] is a proportional function.

∴m-2=0，

Solution: m=2,

Then the equation about x1x-1+1m =1becomes1x-1+12 =1,

Solution: x=3,

Test: substitute x=3 into the simplest common denominator 2(x- 1)=4≠0,

So x=3 is the solution of the original fractional equation,

So the answer is: x = 3.

Comments: This topic mainly examines the understanding of fractional equation and proportional function. The key is to find the value of m, and the basic idea of solving fractional order equation is to "transform ideas" and transform fractional order equation into integral equation. When solving fractional equations, we must pay attention to the test of roots. 2. Test site: the intersection of parabola and X axis; The relationship between roots and coefficients. Special topic: inquiry style. Analysis: (1) First get the values of x 1 and x2 according to the root formula, and then get the sum and product of the two roots.

(2) Substitute the point (-1,-1) into the analytical formula of parabola, and then from d=|x 1-x2|, we can know that D2 = (x1-x2) 2 = (x1+x2). X2=p2, and then from (1) X65438+X2 =-P, x 1? X2=q can lead to a conclusion. Answer: Proof: (1)∵a= 1, b=p, C = Q.

∴△=p2-4q

∴ x =-PP P2-4Q2, that is, x 1=-p+p2-4q2, x2=-p-p2-4q2.

∴x 1+x2=-p+p2-4q2+-p-p2-4q2=-p，

x 1？ x2=-p+p2-4q2？ -p-p2-4q2=q

(2) Substitute (-1,-1) to get p-q=2 and q=p-2.

Let the coordinates where the parabola y=x2+px+q and the X axis intersect at A and B be (x 1, 0) and (x2, 0) respectively.

∫d = | x 1-x2 |，

∴d2=(x 1-x2)2=(x 1+x2)2-4x 1？ x2=p2-4q=p2-4p+8=(p-2)2+4

When p=2, the minimum value of d2 is 4. Comments: This question examines the intersection of parabola and X axis and the relationship between root and coefficient. Knowing that x 1 and x2 are two of the equations x2+px+q=0 is the key to solve this problem. Coordinate and graphic attributes; Pythagorean theorem Analysis: First, make BD⊥x axis pass through point B at D. From A (0 0,2) and B (4 4,3), we can get OA=2, BD=3, OD=4. From the meaning in the question, we can easily prove △AOC∽△BDC, and then we can get OA according to the proportion of corresponding sides in similar triangles.

∫A(0，2)，B(4，3)，

∴OA=2,BD=3,OD=4，

According to the meaning: ∠ACO=∠BCD,

∠∠AOC =∠BDC = 90，

∴△AOC∽△BDC，

∴OA:BD=OC:DC=AC:BC=2:3，

∴OC=25OD=25×4=85，

∴AC=OA2+OC2=24 15，

∴BC=34 15，

∴AC+BC=4 1.

That is, the path length of this beam of light from point A to point B is: 4 1.

So the answer is: 4 1. Comments: This topic examines similar triangles's judgment and nature, Pythagorean theorem and the nature of points and coordinates. The difficulty of this problem is moderate, and the key to solve this problem is to master the method of auxiliary lines and the relationship between incident light and reflected light.