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Bonjour,

Réponse :

[tex]\left \{ {{x^{2} +y^2=34} \atop {xy=15}} \right. \\\\\left \{ {{x^{2} +y^2=34} \atop {y=\frac{15}{x} }} \right. \\\\\left \{ {{x^{2} +(\frac{15}{x})^2=34} \atop {y=\frac{15}{x} }} \right. \\\\\left \{ {{x^{2} +\frac{225}{x^2}=34} \atop {y=\frac{15}{x} }} \right. \\[/tex]

[tex]x^{2} +\frac{225}{x^{2} } =34\\x^{4}-34x^{2} +255 =0\\\underline{\text{On pose }X=x^{2} :}\\\\X^2-34X+255=0\\\Delta=(-34)^2-4\times1\times225=256 > 0\\X_1=\frac{-(-34)-\sqrt{256} }{2\times1} =9\\X_2=\frac{-(-34)+\sqrt{256} }{2\times1} =25\\\\\text{Alors :}\\x^{2} =9 \Longleftrightarrow x=3 \:\:ou\:\:x=-3\\x^{2} =25 \Longleftrightarrow x=5 \:\:ou\:\:x=-5[/tex]

[tex]\underline{\text{Donc :}}\\ \text{Pour }x=3\\xy=15\\3y=15\\\boxed{y=5}\\\\\text{Pour }x=-3\\xy=15\\-3y=15\\\boxed{y=-5}\\\\\text{Pour }x=5\\xy=15\\5y=15\\\boxed{y=3}\\\\\text{Pour }x=-5\\xy=15\\-5y=15\\\boxed{y=-3}[/tex]

[tex]\boxed{\boxed{S=\{(-3;-5);(3;5);(-5;-3),(5;3)\}}}[/tex]

En espérant t'avoir aidé, bonne continuation ! :)