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 =

4qcos^2 (a/2)
2qcos^2(a)
qcos^2 (a/4)

#Algebra #ComplexNumbers

Easy Math Editor

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Comments

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 - 6 years, 3 months ago

Thankyou very much for the solution.

Tarun Singh - 6 years, 3 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$