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.