1. 设圆心为(a, b), 圆半径为r, 圆的方程为: (x-a)^2 + (y-b)^2 = r^2 (1)
圆心(a, b)在直线y=2x上, b = 2a (2)
圆与X轴相切, 切点为(a, 0). 切点在圆上: (a - a)^2 + (0-b)^2 = r^2 (3)
由(3), b^2 = r^2 , b=r 或 b= -r
A. b = r, 由(2), a = r/2
圆C与Y轴的交点的横坐标为0, 代入(1): (0-r/2)^2 +(y - r)^2 = r^2
(y-r)^2 = 3r^2/4
y = (1±√3/2)r
交点的坐标为(0, (1+√3/2)r), (0, (1-√3/2)r)
其距离为(弦长): (1+√3/2)r - (1-√3/2)r = 6
√3r = 6
r = 2√3
b = 2√3; a = √3
圆的方程为: (x - √3)^2 + (y - 2√3)^2 = 12
B. b = -r, 由(2), a = -r/2
与A类似, r = 2√3, b = -2√3; a = -√3
圆的方程为: (x + √3)^2 + (y + 2√3)^2 = 12
2. 设圆心为(a, b), 圆半径为r, 圆的方程为: (x-a)^2 + (y-b)^2 = r^2
圆过原点: a^2 + b^2 = r^2 (1)
圆过点(1, 1): (1- a)^2 + (1-b)^2 = r^2 (2)
由(1)(2): a^2 + b^2 = (1- a)^2 + (1-b)^2
a + b = 1 (3)
圆与X轴的交点的纵坐标为0, 代入圆的方程: (x - a)^2 + (0 - b)^2 = r^2
(x-a)^2 = r^2 - b^2
x = a ± √(r^2 - b^2)
交点的横坐标为: a + √(r^2 - b^2), a - √(r^2 - b^2)
其距离为(弦长): 2√(r^2 - b^2) = 3
r^2 - b^2 = 9/4
r^2 = b^2 + 9/4 (4)
将(4)代入(1): a^2 + b^2 = b^2 + 9/4
a^2 = 9/4
a = ±3/2
A. a = 3/2, b = 1-3/2 = -1/2, r^2 = 1/4 + 9/4 = 5/2
圆的方程为: (x -3/2)^2 + (y + 1/2)^2 = 5/2
B. a = -3/2, b = 1 + 3/2 = 5/2, r^2 = 25/4 + 9/4 = 17/2
圆的方程为: (x + 3/2)^2 + (y - 5/2)^2 = 17/2
3. C1：x²+y²-2x=0, (x-1)^2 + y^2 = 1
圆心为(1, 0), 圆半径为1
直线L：x+√3y=0, y = -x/√3, 斜率为 -1/√3.
设圆C圆心为(a, b), 圆半径为r: (x - a)^2 + (y - b)^2 = r^2 (1)
圆C与直线L：x+√3y=0相切于点P（3,-√3), CP垂直于直线L, 过CP的直线M斜率为 √3. 设M的方程为 y = √3x + c
M过点P（3,-√3), c= -4√3
M的方程为 y = √3x -4√3
M过点C: b = √3a -4√3 = √3(a -4) (2)
C: (a, √3(a -4))
CC1距离为两圆半径之和: (a -1)^2 + [√3(a -4)]^2 = (r+1)^2
(a-1)^2 + 3(a-4)^2 = (r+1)^2 (3)
圆C过点P: (a - 3)^2 + [ (√3(a -4) + √3]^2 = r^2 (4)
(3) -(4)并化简可得: r = 6 - a (5)
(5)代入(3)并化简: a^2 -4a = 0, a(a-4) = 0
a = 0 或 a = 4
A. a = 0, r = 6, b = -4√3
圆C的方程为: x^2 + (y+4√3)^2 = 36
B. a = 4, r = 2, b = 0
圆C的方程为: (x-4)^2 + y^2 = 4