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Byju's Answer
Standard XII
Mathematics
Geometrical Representation of a Complex Number
Indicate the ...
Question
Indicate the point of the complex plane
z
which satisfy the following equation.
Prove that
|
z
1
+
z
2
|
2
+
|
z
1
−
z
2
|
2
=
2
(
|
z
1
|
2
+
|
z
2
|
2
)
for any two complex numbers
z
1
and
z
2
.
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Solution
|
z
1
+
z
2
|
2
=
(
z
1
+
z
2
)
(
¯
z
1
+
¯
z
2
)
⟹
|
z
1
|
2
+
|
z
2
|
2
+
z
2
¯
z
1
+
z
1
¯
z
2
⟹
|
z
1
|
2
+
|
z
2
|
2
+
2
R
e
(
z
1
z
2
)
|
z
1
−
z
2
|
2
=
(
z
1
−
z
2
)
(
¯
z
1
−
¯
z
2
)
⟹
|
z
1
|
2
+
|
z
2
|
2
−
z
2
¯
z
1
−
z
1
¯
z
2
⟹
|
z
1
|
2
+
|
z
2
|
2
−
2
R
e
(
z
1
z
2
)
|
z
1
z
+
z
2
|
2
+
|
z
1
z
−
z
2
|
2
=
2
(
|
z
2
1
|
+
|
z
2
|
2
+
2
R
e
(
z
1
z
2
)
−
2
R
e
(
z
1
z
2
)
)
⟹
|
z
1
z
+
z
2
|
2
+
|
z
1
z
−
z
2
|
2
=
2
(
|
z
2
1
|
+
|
z
2
|
2
)
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0
Similar questions
Q.
For any two non-zero complex numbers
z
1
,
z
2
prove the inequality
(
|
z
1
|
+
|
z
2
|
)
∣
∣
∣
z
1
|
z
1
|
+
z
2
|
z
2
|
∣
∣
∣
≤
2
(
|
z
1
|
+
|
z
2
|
)
Q.
Let
Z
1
=
2
+
3
i
and
z
2
=
3
+
4
i
be two points on the complex plane. Then the set of complex numbers z satisfying
|
z
−
z
1
|
2
+
|
z
−
z
2
|
2
=
|
z
1
−
z
2
|
2
repesents
Q.
For any two complex numbers
z
1
,
z
2
we have
|
z
1
+
z
2
|
2
=
|
z
1
|
2
+
|
z
2
|
2
, then
Q.
If
z
1
and
z
2
are two complex numbers, then prove that
|
z
1
|
+
|
z
2
|
=
∣
∣
∣
z
1
+
z
2
2
+
√
z
1
z
2
∣
∣
∣
+
∣
∣
∣
z
1
+
z
2
2
−
√
z
1
z
2
∣
∣
∣
Q.
If
z
1
=
2
+
3
i
and
z
2
=
3
+
4
i
be two points on the complex plane. Then, the set of complex number
z
is satisfying
|
z
−
z
1
|
2
+
|
z
−
z
2
|
2
=
|
z
1
−
z
2
|
2
represents
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