It is known that there is only one intersection point C between the parabola and the x-axis, and it intersects with the straight line y = x + 2 at two points AB, where a is on the y-axis AC = 2 root 2 (1) to find the analytical formula of the parabola（ It is known that there is only one intersection point C between the parabola and the y-axis, and the parabola and the straight line y = x + 2 intersect at two points AB, where a on the y-axis AC = 2 root sign 2 (1) find the analytical formula of the parabola (2) if point B is on the right side of point a, P is a point on the line AB (point P and ab do not coincide with each other, passing through point P to make the vertical angle of the x-axis parabola at Q. suppose that the length of PQ is MP and the abscissa is x, find the functional relationship between M and x, and write out the value range of the independent variable x (3) Whether there is a point P on the line AB so that the circle with the diameter of the line PQ in 2 passes through the point A. if there is, the coordinates of the point P can be obtained There should be only one intersection point C with the X axis

It is known that there is only one intersection point C between the parabola and the x-axis, and it intersects with the straight line y = x + 2 at two points AB, where a is on the y-axis AC = 2 root 2 (1) to find the analytical formula of the parabola（ It is known that there is only one intersection point C between the parabola and the y-axis, and the parabola and the straight line y = x + 2 intersect at two points AB, where a on the y-axis AC = 2 root sign 2 (1) find the analytical formula of the parabola (2) if point B is on the right side of point a, P is a point on the line AB (point P and ab do not coincide with each other, passing through point P to make the vertical angle of the x-axis parabola at Q. suppose that the length of PQ is MP and the abscissa is x, find the functional relationship between M and x, and write out the value range of the independent variable x (3) Whether there is a point P on the line AB so that the circle with the diameter of the line PQ in 2 passes through the point A. if there is, the coordinates of the point P can be obtained There should be only one intersection point C with the X axis

This problem can be called a heavyweight problem
(1) Find the analytical formula of parabola
∵ the line y = x + 2 intersects the Y axis at point a
The substitution of x = 0 into y = x + 2 results in y = 2
The coordinate of point a is a (0,2), then OA = 2
∵ point C is on the x-axis and AC = 2 √ 2,
In RT △ AOC, from the Pythagorean theorem, we obtain that:
OC party = AC Party -- OA party
=(2 √ 2) - 2
= 4
The coordinate of point C is C (- - 2,0) or C (2,0)
Let the analytic formula of parabola be y = ax + BX + C,
∵ the parabola passes through two points a and C, and the coordinate of point C is (- - 2,0) or C (2,0),
The discussion should be divided into two situations
1. When the parabola passes through a (0,2) and C (- - 2,0), the,
Substituting the coordinates of these two points into y = ax + BX + C, we get:
2 = c --------------------------- ①
0 = 4a -- 2b + c --------------- ②
There is only one intersection point between the parabola and the x-axis,
The discriminant of the root of the quadratic equation AX + BX + C = 0 is 0,
That is: △ = b-square -- 4ac = 0 -------- ③
By solving the equations composed of (1), (2) and (3)
a = 1/2 ,b = 2,c = 2.
The analytical formula of parabola is y = (1 / 2) x + 2x + 2
(at this time, B and C coincide, and B is on the left side of point a)
2. When the parabola passes through a (0,2) and C (2,0), the,
Substituting the coordinates of these two points into y = ax + BX + C, we get:
2 = c --------------------------- ①
0 = 4a + 2b + c --------------- ②
∵ there is only one intersection point C (2,0) between parabola and X axis,
The axis of symmetry of the parabola is x = 2,
That is: - B / (2a) = 2 -------- ③
By solving the equations composed of (1), (2) and (3)
a = 1/2 ,b = -- 2,c = 2.
The analytical formula of parabola is y = (1 / 2) x-2x + 2
(at this time, B and C do not coincide, and B is on the right side of point a)
(pay attention to the two different origins of equation 3)
To sum up, there are two parabola analytic expressions satisfying the meaning of the problem
Y = (1 / 2) x + 2x + 2 or y = (1 / 2) x - 2x + 2
There is another way to find the analytical formula of parabola
There is only one intersection point between the parabola and the x-axis
The parabola in this problem can be regarded as y = ax by left and right translation
The analytic formula of parabola can be set as y = a (x + k) square
Then the two unknowns A and K can be obtained by substituting the coordinates of the two points it passes through
(2) If point B is to the right of point a
From question 1:
In this case, the analytical formula of parabola can only be y = (1 / 2) x-2x + 2
First, find out the abscissa of point B
Where the intersection coordinates, most of the simultaneous equations
From the equations y = x + 2
Y = (1 / 2) x-square -- 2x + 2
The solution is: X1 = 0, X2 = 6
That is: the abscissa of the intersection a of parabola and line is 0, and the abscissa of B is 6
∵ point P is on the line y = x + 2, PQ ⊥ X axis,
The abscissa of point P and Q is X
Substituting x into y = x + 2, the ordinate of point P is (x + 2);
Substituting x into y = (1 / 2) x-2x + 2, we can get
The ordinate of point q is: [(1 / 2) X -- 2x + 2]
The length m of PQ = (x + 2) - [(1 / 2) x-2x + 2]
=-- (1 / 2) x + 3x
The functional relation between M and X is: M = -- (1 / 2) x + 3x
The range of independent variable x is 0 < x < 6
(3) There is a point P on the line ab,
The circle with the diameter of the line segment PQ in (2) can pass through point a,
The coordinates of the point P satisfying the meaning of the question are: P (2,4)
It is known from "the circle with the diameter of line segment PQ passes through point a":
∠ PAQ = 90 ° (the circumferential angle of diameter is 90 °)
In RT △ PAQ, PQ is a hypotenuse,
Through point a, ah ⊥ PQ is set at point H,
Then the length of ah is equal to the abscissa X of point P (or point q)
The abscissa of point h is x and the ordinate is 2,
PH = ordinate of point P -- ordinate of point H
= ( x + 2 ) -- 2
= x
QH = ordinate of point h -- ordinate of point Q
=2 -- [(1 / 2) x-square -- 2x + 2]
=-- (1 / 2) x + 2x
It is easy to prove that RT △ PAH ∽ RT △ AQH
Ψ ah = pH × QH
X = x [-- (1 / 2) x + 2x]
∵ x ＞ 0, divided by X on both sides, the solution is as follows:
X = -- (1 / 2) x + 2x
(1 / 2) x-square -- X = 0
X = 0 or x = 2
The abscissa of point P is 2,
The coordinates of the point P satisfying the meaning of the question are: P (2,4)

It is known that the right intersection point of parabola y = x * 2-2x-3 and X axis is a, and the intersection point of parabola y = x * 2-2x-3 and Y axis is B

First, find out the coordinates of the intersection point with X and Y axes: X axis intersection point: another x = 0, then y = 0 ^ 2-2 * 0-3 = - 3, that is, (0, - 3) y axis intersection point: another y = 0, then 0 = x ^ 2-2x-3, = > (x-3) (x + 1) = 0 = > x = 3 or x = - 1, right intersection point is (3,0), so the straight line passes (0, - 3), (3,0), so the slope of the straight line is k = (0 - (- 3)) / (3-0) = 1

If the parabola y = x ^ 2 - (T + 2) x + 9 has only one intersection with the X axis, then the value of T is ()

If there is only one point of intersection with the x-axis, don't say it is equal to 0
So (T + 2) & sup2; - 36 = 0
t+2=±6
t=-8,t=4

The coordinates of the intersection of the parabola y = x ^ 2 + X + 9 and the axis are

△=b^2-4ac=1-36＜0
The parabola has no intersection with the x-axis
The coordinates of the intersection of the parabola y = x ^ 2 + X + 9 and the Y axis are (0,9)