(1) 设切线斜率为k,方程为y = kx,kx - y = 0
圆P的圆心P为(2r,0),P与 kx - y = 0的距离为圆半径r:
r = |2rk -0|/√(k² + 1)
r²(k²+1) = 4r²k²
3k² = 1
k = ±√3/3
切线方程为y = ±(√3/3)x
(2) 圆方程为(x-2r)² + y² = r²,A(r,0),B(3r,0)
圆P与直线X=5/2不相交,r > 5/2或 3r < 5/2 (r < 5/6)
y=ax^2+bx+c过A,B两点:
ar² + br + c = 0 (1)
9ar² + 3br + c = 0 (2)
(2)-(1):b = -4ar (3)
c = -ar² - br = -ar² + 4ar² = 3ar² (4)
y=ax^2+bx+c的顶点在圆P上,显然顶点为圆上纵坐标最大C(2r,r)或最小处D(2r,-r).
(a) y=ax^2+bx+c过C(2r,r)
r = 4ar² + 2br + c = 4ar² -2r*4ar + 3ar² = - ar²
a = -1/r
(b) y=ax^2+bx+c过C(2r,r)
-r = 4ar² + 2br + c = 4ar² -2r*4ar + 3ar² = - ar²
a = 1/r
直线Y= -ax+c = -ax + 3ar² = 0,x = 3r²,M(3r²,0);
当M在线段PB上运动时2r ≤ 3r² ≤ 3r,2/3 ≤ r ≤ 1
a = -1/r时, -3/2≤ a ≤ -1
a = 1/r 时,1 ≤ a ≤ 3/2