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Ask for advice on a math problem in Wuhan senior high school entrance examination in 2006.
Example 7 (Wuhan, 2006)

It is known that the image of quadratic function y=x2-(m+ 1)x+m intersects with X axis at two points A (x 1 0) and B (x2, 0), the positive semi-axis of Y axis is at point C, and X12+x22 =1x22.

(3) Find the analytic expression of this quadratic function;

(4) Does the straight line passing through point D (0,-) intersect with the image of quadratic function at points M and N, and intersect with X axis at point E, so that points M and N are symmetrical about point E? If it exists, find the resolution function of the straight line MN; If it does not exist, please explain why.

Solution: (1) According to the meaning of the question, yes.

x 1+ x2=m+ 1,x 1x2=m,

By x 12+x22= 10,

That is, (x1+x2) 2-2x1x2 =10,

∴(m+ 1)2-2m= 10,

The solution is m=3 or m=-3,

from△=[-(m+ 1)]2-4m >; 0

Get (m- 1) 2 >: 0, namely m≠ 1,

∵ When x=0, y >, the quadratic function image passes through the positive semi-axis of Y axis at point C, ∴; 0, that is, m>0, so the value of m is 3. So the analytic formula of quadratic function is y=x2-4x+3.

(2) Suppose that a straight line passing through point D(0,-) intersects with the image of quadratic function at points M and N, and an X axis intersects with point E, so that points M and N are symmetrical about point E ... as shown in figure 19.

Let M(xm, ym), N(xn, yn), and the resolution function of the straight line passing through point D(0,-) (which cannot be the Y axis) be y=kx-. Ym+yn=0 when point M and point N are symmetrical about point E.

By y=x2-4x+3.y=kx-

(If y is removed, x2-(4+k)x+ =0,

∴xm+xn=k+4,

∴ym+yn=(kxm- )+(kxn-)

=k(xm+xn)-5

=k(k+4)-5

From ym+yn=0, k2+4k-5=0,

K 1= 1,k2=-5。

When k=-5, the equation x2-(k+4)x+ =0.

Discriminant formula of root of ⊿ < 0, figure 19.

∴K= 1, so the analytical formula of straight line MN is Y=X-

Therefore, the straight line passing through point D(0,-) intersects with the image of quadratic function at points M and N, making points M and N symmetrical about point E. ..

Note: This is a function synthesis problem related to univariate function and bivariate function. Be good at using Vieta theorem to solve problems. It is important to transform point M and point N symmetrically about point E into "YM +Y N=0" in solving problems, but don't forget to check the discriminant ⊿ > 0.