If z = a + ib and |z-2| = |2z -1 |, prove that a ^2 + b ^2 = 1.
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Answer:
Given, a+ib=
2x
2
+1
(x+i)
2
=
2x
2
+1
x
2
+i
2
+2xi
=
2x
2
+1
x
2
−1+i2x
=
2x
2
+1
x
2
−1
+i(
2x
2
+1
2x
)
On comparing real and imaginary parts, we obtain
a=
2x
2
+1
x
2
−1
and b=
2x
2
+1
2x
2
∴a
2
+b
2
=(
2x
2
+1
x
2
−1
)
2
+(
2x
2
+1
2x
)
2
=
(2x
2
+1)
2
x
4
+1−2x
2
+4x
2
=
(2x
2
+1)
2
x
2
+1+2x
2
=
(2x
2
+1)
2
(x
2
+1)
2
∴a
2
+b
2
=
(2x)
2
+1)
2
(x
2
+1)
2
Hence, proved.
Step-by-step explanation:
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