已知动圆P过定点A(-1,0),并且在定圆B:(x-1)^2+y^2=16的内部与其相内切(1)求动圆圆心P的轨迹C的方程(2)过C的右焦点且斜率为k(k≠0)的直线l和C分别交于点M,N,线段MN的垂直平分线和x轴相交于点P(m,0）,

Question

已知动圆P过定点A(-1,0),并且在定圆B:(x-1)^2+y^2=16的内部与其相内切
(1)求动圆圆心P的轨迹C的方程
(2)过C的右焦点且斜率为k(k≠0)的直线l和C分别交于点M,N,线段MN的垂直平分线和x轴相交于点P(m,0）,求实数m的取值范围.

Answer 1

(1)设 C(x,y),半径r,
B(1,0),圆B的半径R = 4
CB = R - r = 4 - r
AC = r,
AC + CB = r + 4 - r = 4
此为以A,B(1,0)为焦点的椭圆,c = 1,a = 2,b² = 4 - 1 = 3
圆心P的轨迹C(椭圆)的方程:x²/4 + y²/3 = 1
(2)直线l:y = k(x - 1)
x²/4 + k²(x - 1)²/3 = 1
(4k² + 3)x² - 8k²x + 4k² - 12 = 0
x₁ + x₂ = 8k²/(4k² + 3)
x₁x₂ = (4k² - 12)/(4k² + 3)
线段MN的中点Q(u,v)
u = (x₁ + x₂)/2 = 4k²/(4k² + 3)
v = (y₁ + y₂)/2 =(kx₁ -k + kx₂ -k)/2 = k(x₁ + x₂ - 2)/(4k² + 3) = -6k/(4k² + 3)
线段MN的垂直平分线斜率为-1/k
方程:y - v = (-1/k)(x - u)
令y = 0,x = m = u + kv
= 4k²/(4k² + 3) -6k²/(4k² + 3)
= -2k²/(4k² + 3)
= -2/(4 + 3/k²)
k趋近于正负无穷大时,m趋近于 -2/4 = -1/2
k趋近于0时,m趋近于0
-1/2 < m < 0