Question Number 194172 by sonukgindia last updated on 29/Jun/23
Commented by Frix last updated on 29/Jun/23
![{ ((x^2 +y^2 =5)),((x^4 +y^4 =55)) :} x=±(√(a−(√b)))∧y=±(√(a+(√b))) { ((2a=5)),((2a^2 +2b=55)) :} a=(5/2)∧b=((85)/4) x=±((√(−10+2(√(85))))/2)i∧y=±((√(10+2(√(85))))/2) x+y=±((√(10+2(√(85))))/2)±((√(−10+2(√(85))))/2)i [4 solutions]](https://www.tinkutara.com/question/Q194175.png)
$$\begin{cases}{{x}^{\mathrm{2}} +{y}^{\mathrm{2}} =\mathrm{5}}\\{{x}^{\mathrm{4}} +{y}^{\mathrm{4}} =\mathrm{55}}\end{cases} \\ $$$${x}=\pm\sqrt{{a}−\sqrt{{b}}}\wedge{y}=\pm\sqrt{{a}+\sqrt{{b}}} \\ $$$$\begin{cases}{\mathrm{2}{a}=\mathrm{5}}\\{\mathrm{2}{a}^{\mathrm{2}} +\mathrm{2}{b}=\mathrm{55}}\end{cases} \\ $$$${a}=\frac{\mathrm{5}}{\mathrm{2}}\wedge{b}=\frac{\mathrm{85}}{\mathrm{4}} \\ $$$${x}=\pm\frac{\sqrt{−\mathrm{10}+\mathrm{2}\sqrt{\mathrm{85}}}}{\mathrm{2}}\mathrm{i}\wedge{y}=\pm\frac{\sqrt{\mathrm{10}+\mathrm{2}\sqrt{\mathrm{85}}}}{\mathrm{2}} \\ $$$${x}+{y}=\pm\frac{\sqrt{\mathrm{10}+\mathrm{2}\sqrt{\mathrm{85}}}}{\mathrm{2}}\pm\frac{\sqrt{−\mathrm{10}+\mathrm{2}\sqrt{\mathrm{85}}}}{\mathrm{2}}\mathrm{i} \\ $$$$\left[\mathrm{4}\:\mathrm{solutions}\right] \\ $$
Answered by cortano12 last updated on 29/Jun/23
$$\:\mathrm{Let}\:\mathrm{x}+\mathrm{y}\:=\:\mathrm{m}\: \\ $$$$\:\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} +\mathrm{2xy}\:=\:\mathrm{m}^{\mathrm{2}} \\ $$$$\:\mathrm{2xy}\:=\:\mathrm{m}^{\mathrm{2}} −\mathrm{5} \\ $$$$\: \\ $$$$\:\left(\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} \right)^{\mathrm{2}} \overset{\:} {\:}=\:\mathrm{25}\: \\ $$$$\:\mathrm{x}^{\mathrm{4}} +\mathrm{y}^{\mathrm{4}} +\mathrm{2}\left(\mathrm{xy}\right)^{\mathrm{2}} =\:\mathrm{25}\: \\ $$$$\:\mathrm{55}+\mathrm{2}\left(\frac{\mathrm{m}^{\mathrm{2}} −\mathrm{5}}{\mathrm{2}}\right)^{\mathrm{2}} =\mathrm{25} \\ $$$$\: \\ $$
Answered by behi834171 last updated on 30/Jun/23
![x^4 +y^4 =(x^2 +y^2 )^2 −2(xy)^2 ⇒55=25−2(xy)^2 ⇒(xy)^2 =−15 (x+y)^2 =x^2 +y^2 −2(xy)=5±2i(√(15)) ⇒x+y=±(√(5±2i(√(15)))) [5+2i(√(15))]^(1/2) =a+ib⇒−40+20i(√(15))=a^2 −b^2 +2iab ⇒ { ((a^2 −b^2 =−40⇒a^2 −(((10(√(15)))/a))^2 =−40⇒)),((2ab=20(√(15)))) :} ⇒a^4 +40a^2 −1500=0 ⇒a^2 =((−40±(√(1600+6000)))/2)=−20±10(√(19)) ⇒a^2 =−(20+43.6),−(20−43.6)=−63.6,23.6 ⇒ { ((a=4.86⇒b=7.96)),((a=7.96i⇒b=−4.86i)) :} ⇒x+y=4.86+7.96i](https://www.tinkutara.com/question/Q194195.png)
$$\boldsymbol{{x}}^{\mathrm{4}} +\boldsymbol{{y}}^{\mathrm{4}} =\left(\boldsymbol{{x}}^{\mathrm{2}} +\boldsymbol{{y}}^{\mathrm{2}} \right)^{\mathrm{2}} −\mathrm{2}\left(\boldsymbol{{xy}}\right)^{\mathrm{2}} \Rightarrow\mathrm{55}=\mathrm{25}−\mathrm{2}\left({xy}\right)^{\mathrm{2}} \\ $$$$\Rightarrow\left({xy}\right)^{\mathrm{2}} =−\mathrm{15} \\ $$$$\left(\boldsymbol{{x}}+\boldsymbol{{y}}\right)^{\mathrm{2}} =\boldsymbol{{x}}^{\mathrm{2}} +\boldsymbol{{y}}^{\mathrm{2}} −\mathrm{2}\left(\boldsymbol{{xy}}\right)=\mathrm{5}\pm\mathrm{2}\boldsymbol{{i}}\sqrt{\mathrm{15}} \\ $$$$\Rightarrow\boldsymbol{{x}}+\boldsymbol{{y}}=\pm\sqrt{\mathrm{5}\pm\mathrm{2}\boldsymbol{{i}}\sqrt{\mathrm{15}}} \\ $$$$\left[\mathrm{5}+\mathrm{2}\boldsymbol{{i}}\sqrt{\mathrm{15}}\right]^{\frac{\mathrm{1}}{\mathrm{2}}} =\boldsymbol{{a}}+\boldsymbol{{ib}}\Rightarrow−\mathrm{40}+\mathrm{20}\boldsymbol{{i}}\sqrt{\mathrm{15}}=\boldsymbol{{a}}^{\mathrm{2}} −\boldsymbol{{b}}^{\mathrm{2}} +\mathrm{2}\boldsymbol{{iab}} \\ $$$$\Rightarrow\begin{cases}{\boldsymbol{{a}}^{\mathrm{2}} −\boldsymbol{{b}}^{\mathrm{2}} =−\mathrm{40}\Rightarrow\boldsymbol{{a}}^{\mathrm{2}} −\left(\frac{\mathrm{10}\sqrt{\mathrm{15}}}{\boldsymbol{{a}}}\right)^{\mathrm{2}} =−\mathrm{40}\Rightarrow}\\{\mathrm{2}\boldsymbol{{ab}}=\mathrm{20}\sqrt{\mathrm{15}}}\end{cases} \\ $$$$\Rightarrow\boldsymbol{{a}}^{\mathrm{4}} +\mathrm{40}\boldsymbol{{a}}^{\mathrm{2}} −\mathrm{1500}=\mathrm{0} \\ $$$$\Rightarrow\boldsymbol{{a}}^{\mathrm{2}} =\frac{−\mathrm{40}\pm\sqrt{\mathrm{1600}+\mathrm{6000}}}{\mathrm{2}}=−\mathrm{20}\pm\mathrm{10}\sqrt{\mathrm{19}} \\ $$$$\Rightarrow\boldsymbol{{a}}^{\mathrm{2}} =−\left(\mathrm{20}+\mathrm{43}.\mathrm{6}\right),−\left(\mathrm{20}−\mathrm{43}.\mathrm{6}\right)=−\mathrm{63}.\mathrm{6},\mathrm{23}.\mathrm{6} \\ $$$$\Rightarrow\begin{cases}{\boldsymbol{{a}}=\mathrm{4}.\mathrm{86}\Rightarrow\boldsymbol{{b}}=\mathrm{7}.\mathrm{96}}\\{\boldsymbol{{a}}=\mathrm{7}.\mathrm{96}\boldsymbol{{i}}\Rightarrow\boldsymbol{{b}}=−\mathrm{4}.\mathrm{86}{i}}\end{cases} \\ $$$$\Rightarrow\boldsymbol{{x}}+\boldsymbol{{y}}=\mathrm{4}.\mathrm{86}+\mathrm{7}.\mathrm{96}\boldsymbol{{i}} \\ $$