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If z1, z2, z3 are the vertices of the ΔABC on the complex plane

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If z1, z2, z3 are the vertices of the ΔABC on the complex plane & are also the roots of the equation z3 – 3αz2 + 3βz + x = 0, then the condition for the ΔABC to be equilateral  triangle is:

 (A) α2= β

(B) α = β2

(C) α = 2β

 (D)α = 3β2

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Best answer

Correct option is (A)

Explation:

Z3 - 3αz2 + 3βz + x=0

z1, z2, z3

⇒ z1+ z2+ z3 =3α

⇒ Z1Z2 + Z2Z3 + Z3Z1 = 3β

⇒ Z1, Z2, Z3 = –x

 (Z1 + Z2 + Z3) 2 = (Z12 + Z2 2 + Z32 + 2(3β)

 (3α)2 = 9β

 α2=β [A]

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