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Byju's Answer
Standard XII
Mathematics
Condition for Coplanarity of Four Points
Let Z1 and ...
Question
Let
Z
1
and
Z
2
be two complex numbers such that
∣
∣
∣
Z
1
−
2
Z
2
2
−
z
1
¯
¯
¯
Z
2
∣
∣
∣
=
1
and
|
Z
2
|
≠
1
, find
|
Z
1
|
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Solution
Given:
∣
∣
∣
Z
1
−
2
Z
2
2
−
Z
1
¯
Z
2
∣
∣
∣
=
1
To find:
|
Z
1
|
On squaring we get
|
Z
1
−
2
Z
2
|
2
=
|
2
−
Z
1
¯
Z
2
|
2
Now we know that
|
Z
|
2
=
Z
¯
Z
⇒
(
Z
1
−
2
Z
2
)
(
¯
Z
1
−
2
¯
Z
2
)
=
(
2
−
Z
1
¯
Z
2
)
(
2
−
¯
Z
1
Z
2
)
⇒
|
Z
1
|
2
−
2
Z
1
¯
Z
2
−
2
Z
2
¯
Z
1
+
4
|
Z
2
|
2
=
4
−
2
¯
Z
1
Z
2
−
2
Z
1
¯
Z
2
+
|
Z
1
|
2
|
Z
2
|
2
⇒
|
Z
1
|
2
−
|
Z
1
|
2
|
Z
2
|
2
=
4
−
4
|
Z
2
|
2
⇒
|
Z
1
|
2
{
1
−
|
Z
2
|
2
}
=
4
{
1
−
|
Z
2
|
2
}
Since
|
Z
2
|
≠
1
⇒
|
Z
1
|
2
=
4
⇒
|
Z
1
|
=
2
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0
Similar questions
Q.
Let
z
1
and
z
2
be two complex numbers such that
∣
∣
∣
z
1
−
2
z
2
2
−
z
1
¯
¯¯¯
¯
z
2
∣
∣
∣
=
1
and
|
z
2
|
≠
1
. Then the value of
|
z
1
|
is
Q.
let
z
1
and
z
2
be two complex number such that
|
1
−
z
1
z
2
|
2
−
|
z
1
−
z
2
|
2
=
k
(
1
−
|
z
1
|
2
)
(
1
−
|
z
2
|
2
)
find the value of
k
Q.
Let
z
1
and
z
2
be two complex numbers such that
∣
∣
∣
z
1
−
2
z
2
2
−
z
1
¯
¯¯¯
¯
z
2
∣
∣
∣
=
1
and
|
z
2
|
≠
1
. Then the value of
|
z
1
|
is
Q.
Let
∣
∣
∣
¯
¯¯¯
¯
z
1
−
2
¯
¯¯¯
¯
z
2
2
−
z
1
¯
¯¯¯
¯
z
2
∣
∣
∣
=
1
and
|
z
2
|
≠
1
,
where
z
1
and
z
2
are complex numbers. Then
|
z
1
|
equals
Q.
Let
Z
1
and
Z
2
be two complex numbers such that
z
1
z
2
+
z
2
z
1
=
1
. Then
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