Question

已知圆C:(x+4)2+y2=4和点a(-2√ 3,0),圆D的圆心在y轴上移动,且与圆C外切,

Answer 1

∠MAN是定值

C(-4,0),设D(0,t)

∵两圆外切

∴圆D半径=CD-圆C半径=√(t^2+16)-2

∴M(0,t+√(t^2+16)-2),N(0,t-√(t^2+16)+2)

∴kAM=[t+√(t^2+16)-2]/2√3,kAN=[t-√(t^2+16)+2]/2√3

MAN为直线AN到AM的角

∴tanMAN=(kAM-kAN)/(1+kAM*kAN)

=[(√(t^2+16)-2)/√3] / {1+[t^2-(√(t^2+16)-2)^2]/12}

=[√(t^2+16)-2]/√3 / {1+(1/12)[t^2-t^2-20+4√(t^2+16)]}

=[√(t^2+16)-2]/√3 / {(1/3)[√(t^2+16)-2]}=√3/3

∴MAN为定值60°