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Byju's Answer
Standard XII
Mathematics
Shortest Distance between Two Skew Lines
Let Z 1 and z...
Question
Let
z
1
and
z
2
be nth roots of unity which subtend a right angle at the origin, then n must be of the form (where, k is an integer)
A
4k+1
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B
4k+2
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C
4k+3
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D
4k
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Solution
The correct option is
D
4k
Since,
a
r
g
z
1
z
2
=
π
2
⇒
z
1
z
2
=
c
o
s
π
2
+
i
s
i
n
π
2
=
i
∴
z
n
1
z
n
2
=
(
i
)
n
⇒
i
n
=
1
⇒
n
=
4
k
Alternate Solution
arg
a
r
g
z
2
z
1
=
π
2
∴
z
2
z
1
=
∣
∣
z
2
z
1
∣
∣
e
i
π
2
⇒
z
2
z
1
=
i
⇒
(
z
2
z
1
)
n
=
i
n
∴
z
1
and
z
2
are nth roots of unity.
z
n
1
=
z
n
2
=
1
⇒
(
z
2
z
1
)
n
=
1
⇒
i
n
=
1
⇒
n
=
4
k
, where k is an integer.
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