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Let the complex numbers z(1),z(2) and z...

Let the complex numbers `z_(1),z_(2)` and `z_(3)` be the vertices of an equailateral triangle. If `z_(0)` is the circumcentre of the triangle , then prove that ` z_(1)^(2) + z_(2)^(2) + z_(3)^(2) = 3z_(0)^(2)`.

A

`z_(0)^(2)`

B

`3 z_(0)^(2)`

C

`z_(0)^(3)`

D

`3 z_(0)^(3)`

Text Solution

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The correct Answer is:
B
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Knowledge Check

  • If z_(1),z_(2),z_(3) are the vertices of an isoscles triangle right angled at z_(2) , then

    A
    `z_(1)^(2)+2z_(2)^(2)+z_(3)^(2)=2z_(2)(z_(1)+z_(3))`
    B
    `z_(1)^(2)+z_(2)^(2)+z_(3)^(2)=2z_(2)(z_(1)+z_(3))`
    C
    `z_(1)^(2)+z_(2)^(2)+2z_(3)^(2)=2z_(2)(z_(1)+z_(3))`
    D
    `2z_(1)^(2)+z_(2)^(2)+z_(3)^(2)=2z_(2)(z_(1)+z_(3))`

  • If z_(1), z_(2), z_(3) are vertices of an equilateral triangle with z_(0) its circumcentre, then z_(1)6(2) + z_(2)^(2) + z_(3)^(2) =

    A
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    B
    `9 z_(0)^(2)`
    C
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    D
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  • If z_(1),z_(2)andz_(3) represent the vertices of an equilateral triangle such that |z_(1)|=|z_(2)|=|z_(3)| , then

    A
    `z_(1)+z_(2)=z_(3)`
    B
    `z_(1)+z_(2)+z_(3)=0`
    C
    `z_(1)z_(2)=(1)/(z_(3))`
    D
    `z_(1)-z_(2)=z_(3)-z_(2)`

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