Multiple choice

The roots of the equation ${ \left( z+\alpha \beta \right) }^{ 3 }={ \alpha }^{ 3 }$ represent the vertices of a triangle, one of whose sides is of length

  1. $\sqrt { 3 } \left| \alpha \beta \right|$
  2. $\sqrt { 3 } \left| \alpha \right|$
  3. $\sqrt { 3 } \left| \beta \right|$
  4. $None\ of\ these$
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B Correct answer
Explanation

The equation (z + alpha*beta)^3 = alpha^3 implies z + alpha*beta = alpha * omega^k, where omega is a cube root of unity. The roots are z = alpha(omega^k - beta). The distance between two roots is |alpha(omega^k1 - omega^k2)|. For k1=0, k2=1, the distance is |alpha(1 - omega)| = |alpha| * sqrt(3).

AI explanation

Taking the cube root of both sides of (z + alpha beta) cubed = alpha cubed gives z + alpha beta = alpha, alpha omega, and alpha omega squared, where omega is a complex cube root of unity. The roots are z1 = alpha(1 - beta), z2 = alpha(omega - beta), and z3 = alpha(omega squared - beta). These three points form an equilateral triangle on the complex plane. The length of any side, for example between z1 and z2, is the magnitude of their difference: |z1 - z2| = |alpha(1 - omega)|. Since the magnitude of 1 - omega is root 3, the side length is root 3 |alpha|.

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