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# Algebra

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Positive real numbers x, y  satisfy the equations x^2 + y^2 = 1 and x^4 + y^4 = 17/19. Find xy.

May 14, 2021

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Positive real numbers $$x$$, $$y$$  satisfy the equations $$x^2 + y^2 = 1$$ and
$$x^4 + y^4 = \dfrac{17}{19}$$.
Find $$xy$$.

$$\begin{array}{|rcll|} \hline \left( x^2 + y^2\right)^2 &=& x^4 + 2x^2y^2 + y^4 \\ \left( x^2 + y^2\right)^2 &=& x^4 + y^4 + 2x^2y^2 \quad | \quad x^2 + y^2 = 1,~ \dfrac{17}{19} \\\\ 1^2 &=& \dfrac{17}{19} + 2x^2y^2 \\\\ 2x^2y^2 &=& 1 - \dfrac{17}{19} \\\\ 2x^2y^2 &=& \dfrac{2}{19} \quad | \quad :2 \\\\ x^2y^2 &=& \dfrac{1}{19} \quad | \quad \text{sqrt both sides} \\\\ \mathbf{ xy } &=& \mathbf{ \dfrac{1}{\sqrt{19}} } \\ \hline \end{array}$$

May 14, 2021