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Question

Question: The roots z<sub>1</sub>, z<sub>2</sub>, z<sub>3</sub> of the equation x<sup>3</sup>+ 3px<sup>2</sup>...

The roots z1, z2, z3 of the equation x3+ 3px2 + 2qx + r = 0 (p, q, r are complex numbers) correspond to points A, B and C. Then triangle ABC is equilateral if –

A

p = q2

B

p2 = 3q

C

p2 = q

D

q2 = 3p

Answer

p2 = q

Explanation

Solution

Sol. For equilateral D

z12+z22+z32z_{1}^{2} + z_{2}^{2} + z_{3}^{2}= z1z2+z2z3+z3z1z_{1}z_{2} + z_{2}z_{3} + z_{3}z_{1}

(z1+z2+z3)2=3(z1z2+z2z3+z3z1)(z_{1} + z_{2} + z_{3})^{2} = 3(z_{1}z_{2} + z_{2}z_{3} + z_{3}z_{1}) (– 3p)2 = 3 × 3q

p2 = q