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Algebra of Complex Numbers

If z 1, z 2, ...

Question

If $z_{1}$ , $z_{2}$ , $z_{3}$ are any three roots of the equation $z^{6} = (z + 1)^{6}$ , then $a r g (\frac{z_{1} - z_{3}}{z_{2} - z_{3}})$ can be equal to

A

$0$

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B

$\frac{π}{4}$

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C

$\frac{π}{6}$

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D

$\frac{π}{2}$

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Solution

The correct option is A
$0$

Roots are collinear.

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Similar questions

z_{1}, z_{2}, z_{3}

are any three roots of the equation

z^{6} = (z + 1)^{6},

arg (\frac{z_{1} - z_{3}}{z_{2} - z_{3}})

can be equal to

z_{1}, z_{2}, z_{3}

are any three roots of the equation

Z^{6} = {(z + 1)}^{6}

a r g (\frac{z_{1} - z_{3}}{z_{2} + z_{3}})

can be equal to-

z_{1}, z_{2}, z_{3}

are the vertices of a triangle in argand plane such that

| z_{1} - z_{2} | = | z_{1} - z_{3} |,

arg (\frac{2 z_{1} - z_{2} - z_{3}}{z_{3} - z_{2}})

z_{1}, z_{2}, z_{3}

be three non-zero complex number, such that

z_{2} \neq z_{1}, a = | z_{1} |, b = | z_{2} | a n d c = | z_{3} |

∣ ∣ ∣ ∣ \begin{matrix} a & b & c b & c & a c & a & b \end{matrix} ∣ ∣ ∣ ∣ = 0

(\frac{z_{3}}{z_{2}})

z_{1}, z_{2}, z_{3}

3

distinct complex numbers such that

\frac{3}{| z_{2} - z_{3} |} = \frac{4}{| z_{3} - z_{1} |} = \frac{5}{| z_{1} - z_{2} |}

, then the value of

\frac{9}{z_{2} - z_{3}} + \frac{16}{z_{3} - z_{1}} + \frac{25}{z_{1} - z_{2}}

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