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Question: If the points A(z), B(–z) and C(z – 1) form an equilateral triangle then value of z is...

If the points A(z), B(–z) and C(z – 1) form an equilateral triangle then value of z is

A

2±3i4\frac{2 \pm \sqrt{3}i}{4}

B

1±3i4\frac{1 \pm \sqrt{3}i}{4}

C

2∓3i4\frac{2 \mp \sqrt{3}i}{4}

D

2±2i4\frac{2 \pm 2i}{4}

Answer

1±3i4\frac{1 \pm \sqrt{3}i}{4}

Explanation

Solution

Sol. If z, – z and z – 1 form equilateral triangle then z2 + z2 + (z – 1)2 = – z2 – z(z – 1) + z(z – 1)

3z2 – 2z + 1 = z2 ̃ 4z2 – 2z + 1 = 0

̃ z = 2±4–168\frac{2 \pm \sqrt{4–16}}{8}

̃ z = 2±23i8\frac{2 \pm 2\sqrt{3}i}{8}̃ z =1±3i4\frac{1 \pm \sqrt{3}i}{4}