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Rotation Concept

Let z 1 and z...

Question

Let $z_{1}$ and $z_{2}$ are the roots of $3 z^{2} + 3 z + b = 0$ . If $O (0), (z_{1}), (z_{2})$ form an equilateral triangle, then $b =$

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Solution

$z_{1} + z_{2} = - 1, z_{1} z_{2} = \frac{b}{3}$
$△ O A B$ is equilateral. So using
$z_{1}^{2} + z_{2}^{2} + z_{3}^{2} = z_{1} z_{2} + z_{2} z_{3} + z_{1} z_{3}$ we get
$0^{2} + z_{1}^{2} + z_{2}^{2} = 0 \times z_{1} + 0 \times z_{2} + z_{1} z_{2}$
$\Rightarrow (z_{1} + z_{2})^{2} - 2 z_{1} z_{2} = z_{1} z_{2}$
$\Rightarrow 1 = 3 z_{1} z_{2}$
$\Rightarrow b = 1$

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Rotation Concept

Standard XII Mathematics

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