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Paragraph: In argand plane consider A ( z ₁ ), B ( z ₂ ) and C ( z ₃ ) as distinct complex numbers lying on the curve | z ^2 |= 3 . Question: If z₁ z^2+z₂ z+z₃=0 and z₂ z^2+z₃ z+z₁=0 each have a common root having modulus unity then ABC is

Options

  1. Aisosceles
  2. Bequilateral
  3. Cisosceles right angled
  4. Dscalene

Correct answer

B. equilateral

Step-by-step solution

Let z_ , z_p be roots of the equation z₁ z^2+z₂ z+z₃=0 such that |z_d |=1 then (z_ +z_ )⁻=z_ z_ ( -z₂ z₁ )⁻= z₃ z₁ z₂^2=z₁ z₃ Hence z₁, z₂, z₃ are in G.P. Since z₁^2+z₂^2+z₃^2=z₁ z₂+z₂ z₃+z₃ z₁ A B C is equilateral. Also PA ^2+ PB ^2+ PC ^2=3 |z₁ |^2- (1+ + ^2 ) z₁^ z - (1+ + ^2 ) z₁ z+ |z₁ |^2+ |z₁ |^2+ |z₁ |^2=6 r^2=18 .

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