x^2 +xy+y^2 =α^2 y^2 +yz+z^2 =β^2 z^2 +zx+x^2 =α^2 +β^2 Find x+y+z for x, y, z ∈R^+


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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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