In the plane rectangular coordinate system xoy, the intersection points of the curve y = x ^ 2-4x + 3 and the two coordinate axes are all on the circle C 1) Find the equation of circle C 2) Is there a real number a such that the circle C and the straight line X-Y + a = 0 intersect at two points a and B, and the angle AOB is 90 degrees

In the plane rectangular coordinate system xoy, the intersection points of the curve y = x ^ 2-4x + 3 and the two coordinate axes are all on the circle C 1) Find the equation of circle C 2) Is there a real number a such that the circle C and the straight line X-Y + a = 0 intersect at two points a and B, and the angle AOB is 90 degrees

1) The intersection points (1,0), (3,0), (0,3) of the curve y = x ^ 2-4x + 3 and the two coordinate axes are all on the circle C,
Let the equation of circle C be (X-2) ^ + (y-b) ^ = R ^
1+b^=r^,
4+(3-b)^=r^,
The subtraction yields 6b-12 = 0, B = 2,
∴r^=5,
The equation of circle C is (X-2) ^ + (Y-2) ^ = 5
(2) Substitute y = x + A, ② into ①, 2x ^ + (2a-8) x + A ^ - 4a-1 = 0,
Let a (x1, Y1), B (X2, Y2), then
x1+x2=4-a,x1x2=(a^-4a-1)/2,
From (2), y1y2 = (x1 + a) (x2 + a) = x1x2 + a (x1 + x2) + A ^,
∠AOB=90°,
0=x1x2+y1y2=a^-4a-1+a(4-a)+a^=a^-1,
A ^ = 1, a = soil 1

In the plane rectangular coordinate system, the intersection of the curve y = x squared + 2x-3 and the coordinate axis is on the circle C Find the equation of circle C. if the circle C and the straight line X-Y + a = O intersect at two points a and B, and OA is perpendicular to ob, find the value of A

By substituting x = 0 and y = 0 into y = x ^ 2 2x-3, we get y = - 3, x ^ 2 2x-3 = 0 to get x 1 = - 3, X2 = 1. Therefore, the coordinates of the intersection point between the curve y = x ^ 2 2x-3 and the coordinate axis are (- 3,0) (1,0) (0, - 3) let the circular equation be (X-B) ^ 2 (y-c) ^ 2 = R ^ 2

In the plane rectangular coordinate system xoy, the circle C1: (x + 3) 2 + (Y-1) 2 = 4 and the circle C2: (x-4) 2 + (Y-5) 2 = 4 are known (1) If the point m ∈ C1 and the point n ∈ C2, find the value range of | Mn |; (2) If the straight line L passes through point a (4, 0), and the chord length cut by circle C1 is 2 3. Find the equation of the straight line L

(1) ∵ the center coordinates of circle C1: (x + 3) 2 + (Y-1) 2 = 4 are (- 3, 1), and the radius is 2. The coordinates of circle C2: (x-4) 2 + (Y-5) 2 = 4 are (4, 5), and the radius is 2,
∴|C1C2|=
65，
Qi
65-4≤|MN|≤
65+4；
(2) Since there is no intersection point between the straight line x = 4 and the circle C1, the slope of the straight line L exists. Let the equation of the line l be y = K (x-4), that is, kx-y-4k = 0,
The distance from the center C1 to the straight line is d = | 7K + 1|
k2+1．
∵ the chord length of the straight line cut by circle C1 is 2
3，
| d = 1, that is, | 7K + 1|
k2+1=1．
The results show that 48k2 + 14K = 0, k = 0, or K = - 7
24．
The linear equation is y = 0, or 7x + 24y-28 = 0

In the plane rectangular coordinate system xoy, the inverse scale function y = K The graph of X and y = 3 The image of X is symmetric about the X axis and intersects with the straight line y = ax + 2 at point a (m, 3). Try to determine the value of A

∵ inverse proportional function y = k
The graph of X and y = 3
The image of X is symmetric about the X axis,
The inverse proportional function y = K
The analytic expression of X is y = - 3
x，
∵ point a (m, 3) in the inverse proportional function y = − 3
In the image of X,
ν M = - 1, that is, the coordinates of point a are (- 1, 3),
∵ point a (- 1,3) is on the line y = ax + 2,
A = - 1 can be obtained
So the value of a is - 1

In the plane rectangular coordinate system xoy, the image of the inverse scale function y = K / X (k is not equal to 0) and the image of y = 3 / X are symmetric about the X axis Since the line y = ax + 2 intersects with point a (m, 3), the value of a is determined

Since y = K / X and y = 3 / X are symmetric about the X axis, k = - 3
If y = 3 is introduced into y = - 3 / x, x = - 1
So a: (- 1,3)
Bring x = - 1, y = 3 into y = ax + 2, 3 = - A + 2
Get a=-1

As shown in the figure, in the plane rectangular coordinate system xoy, the image of the first order function y = KX + B (K ≠ 0) and the inverse scale function y = M The image of X (m ≠ 0) intersects two points a and B in the second and fourth quadrants, and intersects with X axis at point C. the coordinates of point B are (6, n). The line OA = 5, e is a point on the X axis, and sin ∠ AOE = 4 5． (1) The analytic expressions of the inverse proportional function and the first order function are obtained; （2） Find the area of △ AOC

(1) Pass through point a as ad ⊥ X-axis at point D, as shown in the figure, ∵ sin  AOE = 45, OA = 5,  sin ∠ AOE = ADOA = Ad5 = 45,  ad = 4,  do = 52-42 = 3, while point a is in the second quadrant, the coordinates of ⊥ point a are (- 3,4). Substituting a (- 3,4) into y = MX, M = - 12, ﹤ the analytic formula of inverse proportional function is y = - 12x

In the plane rectangular coordinate system xoy, the parabola y = ax 2 + BX + C intersects with the X axis at two points a and B (point a is to the left of point B) In the plane rectangular coordinate system xoy, the parabola y = ax? + BX + C and X axis intersect at two points a and B (point a is on the left side of point B), and intersects with y axis at point C, and the coordinates of point a are (- 3,0). If the straight line y = KX + B passing through the two points of a and C moves down 3 units along the Y axis and passes through the origin exactly, and the symmetry axis of the parabola is a straight line x = - 2

4. In the plane rectangular coordinate system xoy, the parabola y = AX2 + BX + C and X axis intersect at two points a and B (point a is on the left side of point B), and intersects with y axis at point C, and the coordinate of point a is (- 3,0). If the straight line y = KX + B passing through two points of a and C is shifted down 3 units along Y axis, and then passes through the origin exactly, and the symmetry axis of parabola is a straight line x = - 2
(1) Find the function expression of straight line AC and parabola;
(2) If P is a point on the line AC, let the areas of △ ABP and △ BPC be s △ ABP and s △ BPC respectively, and s △ ABP: s △ BPC = 2:3, calculate the coordinates of point P;
(3) If the radius of ⊙ q is 1 and the center of circle Q moves on a parabola, is there any case that ⊙ q is tangent to the coordinate axis in the process of motion? If there is, find out the coordinates of the center Q; if not, please explain the reasons. And explore: if the radius of ⊙ q is r, the center Q moves on the parabola, then when R is taken, ⊙ Q and two sitting axes are tangent at the same time
This question
Some of the 2010 Chengdu test papers have their own to find it,!

As shown in the figure, in the plane rectangular coordinate system, the straight line y = 1 / 2x + 1 and the parabola y = ax? + bx-3 intersect at two points AB, point a is on the X axis, and the ordinate of point B is 3 (2012, Henan) as shown in the figure, in the plane rectangular coordinate system, the straight line y= 12x + 1 and the parabola y = AX2 + bx-3 intersect at two points a and B, point a is on the x-axis, and the ordinate of point B is 3. Point P is a moving point on the parabola below the straight line AB (not coincident with points a and b), passing through point P, the line ab of X axis intersects with point C, and PD ⊥ AB is at point D (1) Find the values of a, B and sin ∠ ACP; (2) Let the abscissa of point p be m ① The length of line segment PD is expressed by algebraic expression containing m, and the maximum value of segment PD length is obtained; ② Connect Pb, segment PC divides △ PDB into two triangles. If there is a suitable value of M, write the value of M directly so that the area ratio of the two triangles is 9:10? If so, write the value of M directly; if not, explain the reason

Refer to the answer to the last question,

It is known that in the plane rectangular coordinate system xoy, the passing point P (0,2) is any straight line intersecting two points with the parabola y = a times the square of X (a > 0), If the intersection point is a and B, then the product of the ordinates of a and B is ()

Let the equation of the straight line be y = KX + 2, and the coordinates of the intersection of the straight line and the parabola a (x1, Y1), B (X2, Y2)
When k = 0, it is easy to know that the ordinate of the intersection point a and B is equal to the ordinate 2 of point P, so the product of the ordinates of a and B is 4
When k ≠ 0, the simultaneous linear equation y = KX + 2, i.e. x = (Y-2) / K and parabolic equation y = ax
By substituting y for X, y = a [(Y-2) / k] 2 is obtained, that is, ay - (4a-k? 2) y + 4A = 0
It is easy to know from the relation between the root and coefficient of quadratic equation (Weida theorem)
y1*y2＝4a/a=4
So the product of the ordinates of a and B is 4

In the plane rectangular coordinate system xoy, it is known that the parabola y = a (x + 1) ^ 2 + C (a > 0) and X axis intersect at two points a and B In the plane rectangular coordinate system xoy, it is known that the parabola y = a (x + 1) ^ 2 + C (a > 0) and X axis intersect at two points a and B (point a is on the left side of point B) and Y axis intersects with point C, and its vertex is m. if the function expression of line MC is y = kx-3, the intersection point with X axis is n, and COS angle BCO = 3 times, the root sign 10 is divided by 10 (1) Find the function expression of the parabola; (2) Is there a point P different from point C on this parabola, so that the triangle with N, P and C as its vertices is a right triangle with NC as a right angle side? If so, find out the coordinates of point p; if not, please explain the reason; (3) If the parabola is moved up and down along its axis of symmetry so that there is always a common point between the parabola and the line NQ, then how many unit lengths can the parabola shift upward and downward? How many unit lengths can the parabola translate downward? I want to draw pictures by myself. I'm in a hurry. I'm in a hurry. I think I can do it. I'll send you the first few questions!

Now I just finished the first question