Find tangent to the x-axis, the center of the circle C is on the line 3x-y = 0, and the chord length of the cut line X-Y = 0 is 2 The equation of a circle of 7

Find tangent to the x-axis, the center of the circle C is on the line 3x-y = 0, and the chord length of the cut line X-Y = 0 is 2 The equation of a circle of 7

If the center of a circle (T, 3T) is tangent to the x-axis, the radius r = 3|t |
∵ distance from center of circle to straight line d = | t − 3T|
2=
2t，
ν from R2 = D2+（
7) 2, t = ± 1
The center of the circle is (1, 3) or (- 1, - 3), and the radius is equal to 3
The equation of circle C is (x + 1) 2 + (y + 3) 2 = 9 or (x-1) 2 + (Y-3) 2 = 9

It is known that the center C of circle C is on the positive half axis of X axis with radius of 5. The chord length of circle C cut by the line X-Y + 3 = 0 is 2 times the root sign 17 (1) Find circular C equation (2) Let the line ax-y + 5 = 0 intersect the circle at two points a and B, and find the value range of real number a (3) Under the condition of (2), is there a real number a such that a and B are symmetric with respect to the straight line L passing through the point (- 2,4)? If so, find the value of the real number a; if not, please explain the reason

First question:
∵⊙ the center of C is on the positive half axis of X axis,

Given that the circle C passes through the point (1,0), and the center of the circle is on the positive half axis of the X axis, the straight line L: y = X-1 is cut by circle C, and the chord length is 2 √ 2, then what is the linear equation passing through the center of the circle and perpendicular to the line l?

If the center coordinate (x0,0) (x0 > 0), then the circle radius = | x0-1|
(x-x0)²+y²=(x0-1)²
Deformation of linear equation: x-y-1 = 0
The distance from the center of a circle to a straight line d = | x0-0-1 | / √ [1? 2 + (- 1)?] = | x0-1 | / √ 2
A right triangle is composed of half of the chord cut by a circle, the radius of the circle, the center of the circle and the vertical line of the straight line
(x0-1)²=(2√2/2)²+[|x0-1|/√2]²
Organize, get
(x0-1)²=4
X0 = - 1 (x0 > 0, omit) or x0 = 3
Center coordinates (3,0)
The slope of the vertical line is the negative reciprocal of the slope of the known straight line and passes through the (3,0) point
Vertical slope = - 1
The linear equation obtained is y-0 = - (x-3), and y = - x + 3

If the circle C passes through the point (1,0), and the center of the circle is on the positive half axis of X, the chord length of the straight line L: cut by the circle is twice the root 2, then the standard equation of circle C is The line L: y = X-1

The center of a circle is a (a, 0), a > 0r = AC = | A-1 | the chord length is 2 √ 2 chord center distance, that is, the distance from a to x-y-1 = 0 d = | a-0-1 | / √ 2 Pythagorean theorem D? + (2 √ 2 / 2) 2 = R? (A-1) Ω / 2 + 2 = (A-1) Ω = 4a-1 = ± 2A > 0A = 3R = | A-1 = 2, so (x-3) 2 + y  = 4

First aid a mathematical problem: find the equation of circle whose center is on the straight line 3x-y = 0, tangent to the x-axis, and cut by the line X-Y = 0 with chord length of (2 Radix 7) (2 roots 7) do not know how to type in mathematical language

Let the center of a circle o be (x, 3x), then its radius is the absolute value of R = 3x, that is, r = [3x]. In the formula of distance from point to straight line, the distance R1 from O to X-Y = 0 is calculated, and R1 square - R square = the square of root 7. Then x can be calculated and solved

It is known that the center of circle C is (2, - 1) and the chord length of the circle cut by the line L: x-y-1 = 0 is 2 2. Find the equation of the circle and the equation of the circle with the smallest area passing through the two ends of the chord

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Given that the circle C is tangent to the Y axis, the center of the circle is on the straight line x-3y = 0, and the chord length cut by the line y = x is 2 root signs 7, and the equation of circle C is solved

Tangent to y-axis
The distance to the y-axis is equal to the radius
(x-a)^2+(y-b)^2=r^2
r=|a|
The center point C is on the line x-3y = 0
a=3b
(x-3b)^2+(y-b)^2=9b^2
String AB = 2 √ 7
The midpoint is D
Then ad = √ 7, AC = r = | 3B|
CD=√(9b^2-7)
Distance from C to y = x = | 3b-b | / √ (1 + 1) = √ (9b ^ 2-7)
b=1,b=-1
(x-3)^2+(y-1)^2=9
(x+3)^2+(y+1)^2=9b^2

A circle is tangent to the y-axis, the chord length cut on the line y = x is 2, the root is 7, and the center of the circle is on the straight line x-3y = 0

Tangent to y-axis
The distance to the y-axis is equal to the radius
(x-a)^2+(y-b)^2=r^2
r=|a|
The center point C is on the line x-3y = 0
a=3b
(x-3b)^2+(y-b)^2=9b^2
String AB = 2 √ 7
The midpoint is D
Then ad = √ 7, AC = r = | 3B|
CD=√(9b^2-7)
Distance from C to y = x = | 3b-b | / √ (1 + 1) = √ (9b ^ 2-7)
b=1,b=-1
So (x-3) ^ 2 + (Y-1) ^ 2 = 9
(x+3)^2+(y+1)^2=9

A circle is tangent to the y-axis, its center is on the line x-3y = 0, and the line y = x cuts the circle, the chord length is 2 times the root sign 7, and the equation of the circle is solved

Tangent to y-axis
The distance to the y-axis is equal to the radius
(x-a)^2+(y-b)^2=r^2
r=|a|
The center point C is on the line x-3y = 0
a=3b
(x-3b)^2+(y-b)^2=9b^2
String AB = 2 √ 7
The midpoint is D
Then ad = √ 7, AC = r = | 3B|
CD=√(9b^2-7)
Distance from C to y = x = | 3b-b | / √ (1 + 1) = √ (9b ^ 2-7)
b=1,b=-1
So (x-3) ^ 2 + (Y-1) ^ 2 = 9
(x+3)^2+(y+1)^2=9

The chord length of a line passing through the origin and with an inclination angle of 60 ° cut by the circle x2 + y2-4y = 0 is () A. Three B. 2 C. Six D. 2 Three

The equation of circle x2 + y2-4y = 0 can be transformed into:
x2+（y-2）2=4，
That is, the center of the circle is a (0, 2), and the radius is r = 2,
The distance from a to the straight line on, that is, the chord center distance is 1,
∴ON=
3，
The chord length is 2
3，
Therefore, D