已知复数z=(-1+3i)(1-i)/i-(1+3i)/i,w=z+ai,a属于R.当Iw/zI小于等于根号2,求a的取值范围

题目

已知复数z=(-1+3i)(1-i)/i-(1+3i)/i,w=z+ai,a属于R.当Iw/zI小于等于根号2,求a的取值范围

答案

z=(-1+3i)(1-i)/i-(1+3i)/i=(-1+3i-i+3)/i-(1+3i)/i=(1-i)/i=-i-1;
w=(a-1)i-1;
|w/z|=|(a-1)i-1|/|-i-1|≤√2;
√[(a-1)^2+1]≤2
0≤a^2-2a+2≤4;a属于R.
得1-√3≤a≤1+√3