"Let z1, z2 be the roots of the equation z2 + az + 12 = 0 and z1, z2 form an equilateral triangle with origin. Then, the value of |a| is :"

Question

Accepted Answer

"For equilateral triangle with vertices z₁, z₂ and z₃,

$z_1^2 + z_2^2 + z_3^3 = {z_1}{z_2} + {z_2}{z_3} + {z_3}{z_1}$

Here one vertex z₃ is 0

$\therefore$ $z_1^2 + z_2^2 = {z_1}{z_2} + 0 + 0$

Given, z₁, z₂ are roots of ${z^2} + az + 12 = 0$

$\therefore$ ${z_1} + {z_2} = - a$

${z_1}{z_2} = 12$

$\therefore$ $z_1^2 + z_2^2 + 2{z_1}{z_2} = {z_1}{z_2} + 2{z_1}{z_2}$

$\Rightarrow {({z_1} + {z_2})^2} = 3{z_1}{z_2}$

$\Rightarrow {( - a)^2} = 3 \times 12$

$\Rightarrow {a^2} = 36$

$\Rightarrow a = \pm 6$

$\Rightarrow |a|\, = 6$"

Let z₁, z₂ be the roots of the equation z² + az + 12 = 0 and z₁, z₂ form an equilateral triangle with origin. Then, the value of |a| is :

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Let z1, z2 be the roots of the equation z2 + az + 12 = 0 and z1, z2 form an equilateral triangle with origin. Then, the value of |a| is :

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Let z₁, z₂ be the roots of the equation z² + az + 12 = 0 and z₁, z₂ form an equilateral triangle with origin. Then, the value of |a| is :