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Need help in complex numbers

Hello, I am not able to solve this question so can anyone please help me. The question as it is -
Let z1 and z2 be the roots of the equation z^2 + pz + q = 0, where p and q may be complex number. Let A and B represent z1 and z2 in the complex plane. If angle AOB=a not equal to 0 and OA=OB,where O is the origin then p^2 =

  1. 4qcos^2 (a/2)
  2. 2qcos^2(a)
  3. qcos^2 (a/4)

Note by Tarun Singh
2 years, 4 months ago

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Answer is 1.

\(By\quad rotation\quad ,\\ { z }_{ 1 }={ z }_{ 2 }{ e }^{ ia }\\ From\quad quadratic\quad ,\\ { z }_{ 1 }{ z }_{ 2 }=q\\ Eliminating\quad we\quad get\quad ,\\ { z }_{ 1 }=\sqrt { q } { e }^{ -\cfrac { ia }{ 2 } }\\ { z }_{ 2 }=\sqrt { q } { e }^{ \frac { ia }{ 2 } }\\ Sum\quad of\quad roots\quad is\quad p,\\ Hence\quad ,\\ { p }^{ 2 }=4q\cos ^{ 2 }{ \frac { a }{ 2 } } \) Rohit Shah · 2 years, 4 months ago

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@Rohit Shah Thankyou very much for the solution. Tarun Singh · 2 years, 4 months ago

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