求满足这两个式子的复数arg(z-2)=3/4π,|z+2|=3
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![求满足这两个式子的复数arg(z-2)=3/4π,|z+2|=3](/uploads/image/z/2906183-47-3.jpg?t=%E6%B1%82%E6%BB%A1%E8%B6%B3%E8%BF%99%E4%B8%A4%E4%B8%AA%E5%BC%8F%E5%AD%90%E7%9A%84%E5%A4%8D%E6%95%B0arg%EF%BC%88z-2%EF%BC%89%3D3%2F4%CF%80%2C%7Cz%2B2%7C%3D3)
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求满足这两个式子的复数arg(z-2)=3/4π,|z+2|=3
求满足这两个式子的复数arg(z-2)=3/4π,|z+2|=3
求满足这两个式子的复数arg(z-2)=3/4π,|z+2|=3
答:
arg(z-2)=3π/4,复数z-2的辐角为3π/4
|z+2|=3
设z=x+yi
则z-2=x-2+yi
z+2=x+2+yi
依据题意有:
cos(3π/4)=(x-2)/√[(x-2)^2+y^2]=-√2/2
sin(3π/4)=y / √[(x-2)^2+y^2]=√2/2,y>0
√[(x+2)^2+y^2]=3
所以:
2(x-2)^2=(x-2)^2+y^2,y^2=(x-2)^2
(x+2)^2+y^2=9
解得:x^2=1/2,x=√2/2或者x=-√2/2
x=√2/2时,y=2-√2/2
x=-√2/2时,y=2+√2/2
所以:
z=√2/2+(2-√2/2)i
或者z=-√2/2+(2+√2)i