- 复数问题复数Z满足4<=Z+16/Z<=10,求(1
- 复数Z满足4<=Z+16/Z<=10,
求(1)|Z|的取值范围,
(2)复数Z对应点P的轨迹
- 设Z=X+iY,其中X、Y是实数,则
Z+16/Z=(X+iY)+16/(X+iY)=(X+iY)+16(X-iY)/(X^2+Y^2)
=[X+16X/(X^2+Y^2)]+i[Y-16Y/(X^2+Y^2)]
由4<=Z+16/Z<=10知,Z+16/Z是实数,所以
Y-16Y/(X^2+Y^2)=0 ==> Y=0 或 X^2+Y^2=16
按题目的意思应该Z不是实数,所以Y=0舍去,∴X^2+Y^2=16
Z+16/Z=X+16X/(X^2+Y^2)=2X
(1)|Z|=√(X^2+Y^2)=√16=4
(2)由4<=Z+16/Z<=10 ==> 4<=2X<=10 ==> 2<=X<=5
所以复数Z对应点P的轨迹是:圆X^2+Y^2=16在直线X=2的右边部分,用复数形式表示是:圆|Z|=4(Re(Z)>=2)
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如果题目里Z可以是实数,则上面解答中Y=0不要舍去,此时
4<=Z+16/Z<=10就是4<=X+16/X<=10,其中X>0,解不等式组
X+16/X>=4 ==> X^2-4X+16>=0 ==> 对一切X>0
X+16/X<=10 ==> X^2-10X+16=0 ==> 2<=X<=8
不等式组的解是:2<=X<=8
(1)|Z|=|X|,所以2<=|Z|<=8;
(2)Z的轨迹是Y=0 (2<=X<=8),用复数形式表示为:
Im(Z)=0 (2<=Re(Z)<=8)
上面两种情形综合起来就是最完整的解答,即
(1)2<=|Z|<=8;
(2)轨迹是圆X^2+Y^2=16在直线X=2的右边部分与X轴上从X=2到X=8的线段。