Q.2 analytic geometry straight line and circle, hope to answer as soon as possible, 1. It is known that a circle centered on point C (T, 2 / T) (t ∈ R, t ≠ 0) intersects with X-axis at point O and point a, and intersects with Y-axis at point O and point B, where point O is the origin of rectangular coordinate system. Let y = - 2X + 4 intersect with circle C at point m, N, and if om = on, find the equation of circle C 2. Let the image of the quadratic function f (x) = x & # 178; + 2x + B (x ∈ R) have three intersections with the two coordinate axes in the plane rectangular coordinate system xoy, and the circle passing through the three intersections is marked as C (1) I have worked out the value range of B

Q.2 analytic geometry straight line and circle, hope to answer as soon as possible, 1. It is known that a circle centered on point C (T, 2 / T) (t ∈ R, t ≠ 0) intersects with X-axis at point O and point a, and intersects with Y-axis at point O and point B, where point O is the origin of rectangular coordinate system. Let y = - 2X + 4 intersect with circle C at point m, N, and if om = on, find the equation of circle C 2. Let the image of the quadratic function f (x) = x & # 178; + 2x + B (x ∈ R) have three intersections with the two coordinate axes in the plane rectangular coordinate system xoy, and the circle passing through the three intersections is marked as C (1) I have worked out the value range of B

1. Idea: Mn is perpendicular to OC, so we can use the slope to find out the value of T, and then the center and radius OC of circle C can be obtained; 2. Idea: (1) the value range of B; idea: combining number with shape, symmetry axis X = - 1, we find that there are three intersections with two coordinate axes In a word, you are right. (2) find the equation of circle C

Given the circle C1: x2 + y2-6y = 0, C2: (X-2 √ 3) 2 + (Y-1) 2 = 1 (1), prove the circumtangent of two circles, and the x-axis is their outer line
Given the circle C1: x2 + y2-6y = 0, C2: (X-2 √ 3) 2 + (Y-1) 2 = 1 (1), prove the circumtangent of two circles, and the X axis is one of their common tangents. (2) find out their other common tangent equation

（1）
Two circles C1: x ^ 2 + (Y-3) ^ 2 = 9, C2: (X-2 √ 3) ^ 2 + (Y-1) ^ 2 = 1
Distance between center C1 and C2 = √ [(2 √ 3) ^ 2 + (3-1) ^ 2] = 4 = R1 + R2
Two circles are tangent
The distance between center C1 and X axis = 3 = R1, the distance between center C2 and X axis = 1 = R2
The x-axis is a common tangent to them
（2）
Let the tangent equation of circle C1 be ax + B (Y-3) - 3 = 0
The tangent equation of circle C2 is a (X-2 √ 3) + B (Y-1) - 1 = 0
Lianlide B = √ 3a-1
Because the line between the center C1 and C2 is the bisector of the angle of the common tangent
So K2 = 2K = 2 * (1-3) / (2 √ 3) = - 2 √ 3 / 3 = - A / b
a=2√3/3,b=1
The other equation is 2 √ 3A + 3y-18 = 0

Two known circles C1: x ^ 2 + y ^ 2 = 1 and C2: (X-2) ^ 2 + (Y-2) ^ 2 = 5
Find the linear equation of equal chord length which passes through point P (0,1) and is cut by two circles

Let the linear equation be y = KC + B, C1 and C2 intersect at the point (0,1), and the line passes through the point (0,1), then the linear equation can be written as y = KX + 1, and the line determined by the midpoint (1,1) and (0,1) of point (0,0) and (2,2) is perpendicular to the line, K1 = (1-1) / (0-1) = 0, so the slope of the line does not exist. Then the linear equation is x = 0

Given that X and y are real numbers, and the root sign X-2 plus y ^ 2 plus 6y plus 9 equals 0, find the value of (x + y) ^ 2013

√(x-2)+(y+3)²=0
If one is greater than 0, the other is less than 0
So both are equal to zero
So X-2 = 0, y + 3 = 0
x=2,y=-3
x+y=-1
(x+y)^2013=-1

Calculate {the square of 2 (X-Y) - the square of 8 (X-Y) + the square of 6y (X-Y)} the square of 2 (X-Y) is equal to

【2（x-y）²-8（x-y)²（x+y）+6y（x-y）²】÷2（x-y）²
=2（x-y）²÷2(x-y)²-8（x-y)²（x+y）÷2(x-y)²+6y（x-y）²÷2(x-y)²
=1-4(x+y)+3y
=1-4x-4y+3y
=1-4x-y

Is it OK to use the rotational symmetry only if the integral region satisfies the rotational symmetry? What conditions should the integrand satisfy?

In short, the rotational symmetry property,
x. Y interchanges, D remains unchanged

5. The definition of rotation symmetry of function is given
5. Give the definition of function rotation symmetry
(1) The definition of symmetry is given
(2) The definition of rotation is given

Symmetric formula: any two variables are exchanged and the analytic formula is unchanged, such as a + B + C, AB + BC + Ca, AAB + ABB + AAC + ACC + BBC + BCC, etc
Rotation symmetry formula: the formula that transforms all variables in order (such as a → B, B → C, C → a), and the analytic formula remains unchanged, such as
AAB + BBC + CCA, etc
Note that the symmetry must be rotation symmetry, and rotation symmetry is not necessarily symmetry, for example, AAB + BBC + CCA, exchange a, B, get ABB + BCC + CAA, is no longer the original, so AAB + BBC + CCA is rotation symmetry, not symmetry

Is there any condition for using the rotational symmetry of double integral
Should the integral domain and integrand satisfy the rotational symmetry, or just one of them?
In addition, for example, what does the symmetry of the double integral region mean? Is it symmetrical to both the X and Y axes?

Using the symmetry of double integral to solve the problem requires the symmetry of integral region and function
For example, if the integral domain is symmetric about the X axis
If the integrand is an odd function of Y, then the double integral is 0
If it is an even function of Y, then it is equal to 2 ∫ (D1) f (x, y) DXDY, D1 is half of the region~

What are the conditions for using the rotational symmetry of the integral domain?
Is the integral domain X and Y interchangeable and invariant? Can the absolute value of integrand be used
Let's talk about the double integral. Let's give a brief introduction. It's better to give an example,

The rotation symmetry of coordinates is simply to rename the coordinate axis. If the expression of the function in the integral interval remains unchanged, then the integral values of X, y, Z in the integrand will remain unchanged after the same change. (1) for surface integral, the integral surface is u (x, y, z) = 0. If x, y, Z in the function U (x, y, z) = 0 is replaced by Y, Z, x, then the integral value of the integrand will remain unchanged

What is the symmetry theorem of double integral?

When the function of two variables is continuous, the integration has nothing to do with the order of integration of X and Y. the first integration of X and the first integration of Y are the same