已知z、w、x为复数,且x=(1+3i)•z,w=z2+i且|w|=52.(1)若w为大于0的实数,求复数x.(2)若x为纯虚数,求复数w.-高二数学

题目简介

已知z、w、x为复数,且x=(1+3i)•z,w=z2+i且|w|=52.(1)若w为大于0的实数,求复数x.(2)若x为纯虚数,求复数w.-高二数学

题目详情

已知z、w、x为复数,且x=(1+3i)•z,w=
z
2+i
且|w|=5
2

(1)若w为大于0的实数,求复数x.
(2)若x为纯虚数,求复数w.
题型:解答题难度:中档来源:不详

答案

(1)∵x=(1+3i)•z,∴z=class="stub"x
1+3i

若w为大于0的实数,
∵w=class="stub"z
2+i
=class="stub"x
(1+3i)(2+i)
=class="stub"x
-1+7i
,|w|=5
2

则有 5
2
=class="stub"x
-1+7i
,∴x=-5
2
+35
2
i.
(2)若x为纯虚数,设x=bi,b≠0.
由(1)可得 |class="stub"x
-1+7i
|
=|class="stub"bi
-1+7i
|
=5
2
,∴b=±50.
∴w=class="stub"x
-1+7i
=class="stub"50i
-1+7i
=7-i,或w=class="stub"x
-1+7i
=class="stub"-50i
-1+7i
=-7+i.

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