If z1=2-i and z2=-2+i, find (i) Im(z1z2/conjugate of z1) (ii) Re(1/z1 x conjugate of z1) (iii) Im(1/z1 x conjugate of z1)
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Step-by-step explanation:
Given:
z
1
=2−i⇒
z
1
ˉ
=2+i,z
2
=−2+i
(i) z
1
z
2
=(2−i)(−2+i)=−4+2i+2i−i
2
=−4+4i−(−1)=−3+4i
∴
z
1
ˉ
z
1
z
2
=
2+i
−3+4i
=
(2+i)(2−i)
(−3+4i)(2−i)
=
2
2
+1
2
−6+3i+8i−4i
2
=
2
2
+1
2
−6+11i−4(−1)
=
5
−2+11i
=
5
−2
+
5
11
i
∴Re(
z
1
ˉ
z
1
z
2
)=
5
−2
(ii)
z
1
z
1
ˉ
1
=
(2−i)(2+i)
1
=
(2)
2
+(1)
2
1
=
5
1
∴Im(
z
1
z
1
ˉ
1
)=0
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