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Let z(1),z(2) and z(3) be three complex ...

Let `z_(1),z_(2)` and `z_(3)` be three complex number such that `|z_(1)-1|= |z_(2) - 1| = |z_(3) -1| and arg ((z_(3) - z_(1))/(z_(2) -z_(1))) = (pi)/(6)`
then prove that `z_(2)^(3) + z_(3)^(3) + 1 = z_(2) + z_(3) + z_(2)z_(3)`.

Text Solution

Verified by Experts

Clearly, `z_(1),z_(2)` and `z_(3)` lie on a circle having center at `(1,0).`
Also, `angle BAC=30^(@)`

So, `DeltaBOC` is equilateral.
`impliesz_(2)^(2)+z_(3)^(3)+1=z_(2)+z_(3)+z_(2)z_(3)`
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