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Byju's Answer
Standard XII
Mathematics
Complex Numbers
If z1,z2∈ C...
Question
If
z
1
,
z
2
∈
C
1
then prove that :
|
z
1
−
z
2
|
≥
|
z
z
|
−
|
z
2
|
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Solution
Let the sides of triangle are
z
1
,
z
2
,
z
1
−
z
2
difference of two sides is less than third side
|
z
1
|
−
|
z
2
|
≤
|
z
1
−
z
2
|
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0
Similar questions
Q.
If
z
1
and
z
2
are two complex numbers, then prove that
|
z
1
|
+
|
z
2
|
=
∣
∣
∣
z
1
+
z
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2
+
√
z
1
z
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∣
∣
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−
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Q.
(a) For any two non-zero complex numbers
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and
z
2
if
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z
1
+
z
2
|
=
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z
1
|
+
|
z
2
|
, then prove that
a
r
g
z
1
−
a
r
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is zero.
(b) Prove the above result if we have
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z
1
−
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2
|
=
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z
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|
−
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Q.
Prove that
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z
1
|
+
|
z
2
|
=
∣
∣
∣
1
2
(
z
1
+
z
2
)
+
√
z
1
z
2
∣
∣
∣
+
∣
∣
∣
1
2
(
z
1
+
z
2
)
−
√
z
1
z
2
∣
∣
∣
.
Q.
If
z
1
and
z
2
are complex numbers and
u
=
√
z
1
z
2
, then prove that
|
z
1
|
+
|
z
2
|
=
∣
∣
∣
z
1
+
z
2
2
+
u
∣
∣
∣
+
∣
∣
∣
z
1
+
z
2
2
−
u
∣
∣
∣
.
Q.
If for the complex numbers
z
1
,
z
2
,
.
.
.
.
.
,
z
n
,
|
z
1
|
=
|
z
2
|
=
.
.
.
.
.
=
|
z
n
|
=
1
. Then prove that
|
¯
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
¯
z
1
+
z
2
+
.
.
.
.
.
+
z
n
|
=
∣
∣
∣
1
z
1
+
1
z
2
+
.
.
.
.
.
.
+
1
z
n
∣
∣
∣
.
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