Complex NumbersHard
Question
If 1, α1, α2, α3, .....α8, are nine, ninth roots of unity (taken in counter-clockwise sequence) then |(2 - α1) (2 - α3) (2 - α5) (2 - α7)| is equal to
Options
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Solution
(x - 1) (x - α1) (x - α2) (x - α87) = x9 - 1
∴ (2 - α1) (2 - α2)........(2 - α8) = 29 - 1
Now since 2 - α1 and 2 - α8 are conjugates of each other
∴ |2 - α1| = |2 - α8|
similarly
|2 - α2| = |2 - α7|,
|2 - α3| = |2 - α6| and |2 - α4| = |2 - α5|
∴ |(2 - α1) (2 - α3) (2 - α5) (2 - α7)| =
∴ (2 - α1) (2 - α2)........(2 - α8) = 29 - 1
Now since 2 - α1 and 2 - α8 are conjugates of each other
∴ |2 - α1| = |2 - α8|
similarly
|2 - α2| = |2 - α7|,
|2 - α3| = |2 - α6| and |2 - α4| = |2 - α5|
∴ |(2 - α1) (2 - α3) (2 - α5) (2 - α7)| =
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