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Question Number 209694 by Frix last updated on 18/Jul/24

x^{2} + x y + y^{2} = α^{2}

y^{2} + y z + z^{2} = β^{2}

z^{2} + z x + x^{2} = α^{2} + β^{2}

Find x + y + z for x, y, z \in R^{+}

Answered by mr W last updated on 18/Jul/24

Commented by mr W last updated on 18/Jul/24

\frac{1}{2} (x y + y z + z x) \times \frac{\sqrt{3}}{2} = \frac{α β}{2}

\Rightarrow x y + y z + z x = \frac{2 α β}{\sqrt{3}}

2 (x^{2} + y^{2} + z^{2}) + x y + y z + z x = 2 (α^{2} + β^{2})

{(x + y + z)}^{2} - \frac{3}{2} (x y + y z + z x) = α^{2} + β^{2}

{(x + y + z)}^{2} = α^{2} + β^{2} + \sqrt{3} α β

\Rightarrow x + y + z = \sqrt{α^{2} + β^{2} + \sqrt{3} α β}

Commented by Frix last updated on 18/Jul/24

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