find all non zero complex numbers z sqtisfying z = iz2
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Answer:
We know that
z
z
ˉ
=∣z∣
2
Given:
z
ˉ
=iz
2
...(1)
Taking mod on both sides, we get
∣z∣=∣z∣
2
⇒∣z∣=0 or ∣z∣=1
It is given the z is a non-zero number, therefore ∣z∣=1
Now,
Multiply (1) both sides with with z, we get
z
z
ˉ
=iz
3
⇒∣z∣
2
=iz
3
⇒z
3
=−i
⇒z
3
−i
3
=0
⇒(z−i)(z
2
+iz−1)=0
Hence
z=i and z
2
+iz−1=0
z
2
+iz−1=0 Gives
z=
2
−i±
−1+4
z=
2
−i±
3
So z can be (0,1) or (±
2
3
,−
2
1
)
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