Question Number 210958 by RojaTaniya last updated on 24/Aug/24
Commented by Ghisom last updated on 26/Aug/24
$$\mathrm{I}\:\mathrm{get} \\ $$$${x}=\frac{\mathrm{36}}{\mathrm{25}}\wedge{y}=\frac{\mathrm{64}}{\mathrm{25}} \\ $$$${x}=−\frac{\mathrm{7}}{\mathrm{16}}+\frac{\mathrm{3}}{\mathrm{2}}\mathrm{i}\wedge{y}=\frac{\mathrm{63}}{\mathrm{400}}−\frac{\mathrm{27}}{\mathrm{50}}\mathrm{i} \\ $$$${x}=−\frac{\mathrm{7}}{\mathrm{16}}−\frac{\mathrm{3}}{\mathrm{2}}\mathrm{i}\wedge{y}=\frac{\mathrm{63}}{\mathrm{400}}+\frac{\mathrm{27}}{\mathrm{50}}\mathrm{i} \\ $$
Commented by RojaTaniya last updated on 26/Aug/24
$${kindly}\:{provoide}\:{details}\:{solution}. \\ $$
Answered by Ghisom last updated on 26/Aug/24
![(1/(x+y))=t (√x)(1+t)=(6/4) ⇒ t=((3−2(√x))/(2(√x))) (√y)(1−t)^( (∗)) =(6/5) ⇒ t=((5(√y)−6)/(5(√y))) _(for x, y ∈R)^((∗) ⇒ t<1) ((5(√y)−6)/(5(√y)))=((3−2(√x))/(2(√x))) ⇒ y=((144x)/(25(4(√x)−3)^2 )) insert & transform (√x)(1+(1/(x+y)))−(3/2)=0 x^(1/2) (x^2 −3x^(3/2) +((1669)/(400))x−((2307)/(800))x^(1/2) +(9/(16)))=0 obviously x≠0 x^2 −3x^(3/2) +((1669)/(400))x−((2307)/(800))x^(1/2) +(9/(16)) ((√x)−(6/5))((√x)−(3/(20)))(x−(3/2)(√x)+((25)/(16)))=0 ⇒ x=((36)/(25))∧y=((64)/(25)) x=(9/(400))∧y=(4/(25)) _(⇒ false)^([∗] t=4) x=−(7/(16))±(3/2)i∧y=((63)/(400))∓((27)/(50))i](https://www.tinkutara.com/question/Q211039.png)
$$\frac{\mathrm{1}}{{x}+{y}}={t} \\ $$$$\sqrt{{x}}\left(\mathrm{1}+{t}\right)=\frac{\mathrm{6}}{\mathrm{4}}\:\Rightarrow\:{t}=\frac{\mathrm{3}−\mathrm{2}\sqrt{{x}}}{\mathrm{2}\sqrt{{x}}} \\ $$$$\sqrt{{y}}\left(\mathrm{1}−{t}\overset{\:\left(\ast\right)} {\right)}=\frac{\mathrm{6}}{\mathrm{5}}\:\Rightarrow\:{t}=\frac{\mathrm{5}\sqrt{{y}}−\mathrm{6}}{\mathrm{5}\sqrt{{y}}}\:\:\:\:\:_{\mathrm{for}\:{x},\:{y}\:\in\mathbb{R}} ^{\left(\ast\right)\:\Rightarrow\:{t}<\mathrm{1}} \\ $$$$\frac{\mathrm{5}\sqrt{{y}}−\mathrm{6}}{\mathrm{5}\sqrt{{y}}}=\frac{\mathrm{3}−\mathrm{2}\sqrt{{x}}}{\mathrm{2}\sqrt{{x}}}\:\Rightarrow\:{y}=\frac{\mathrm{144}{x}}{\mathrm{25}\left(\mathrm{4}\sqrt{{x}}−\mathrm{3}\right)^{\mathrm{2}} } \\ $$$$\mathrm{insert}\:\&\:\mathrm{transform} \\ $$$$\sqrt{{x}}\left(\mathrm{1}+\frac{\mathrm{1}}{{x}+{y}}\right)−\frac{\mathrm{3}}{\mathrm{2}}=\mathrm{0} \\ $$$${x}^{\mathrm{1}/\mathrm{2}} \left({x}^{\mathrm{2}} −\mathrm{3}{x}^{\mathrm{3}/\mathrm{2}} +\frac{\mathrm{1669}}{\mathrm{400}}{x}−\frac{\mathrm{2307}}{\mathrm{800}}{x}^{\mathrm{1}/\mathrm{2}} +\frac{\mathrm{9}}{\mathrm{16}}\right)=\mathrm{0} \\ $$$$\mathrm{obviously}\:{x}\neq\mathrm{0} \\ $$$${x}^{\mathrm{2}} −\mathrm{3}{x}^{\mathrm{3}/\mathrm{2}} +\frac{\mathrm{1669}}{\mathrm{400}}{x}−\frac{\mathrm{2307}}{\mathrm{800}}{x}^{\mathrm{1}/\mathrm{2}} +\frac{\mathrm{9}}{\mathrm{16}} \\ $$$$\left(\sqrt{{x}}−\frac{\mathrm{6}}{\mathrm{5}}\right)\left(\sqrt{{x}}−\frac{\mathrm{3}}{\mathrm{20}}\right)\left({x}−\frac{\mathrm{3}}{\mathrm{2}}\sqrt{{x}}+\frac{\mathrm{25}}{\mathrm{16}}\right)=\mathrm{0} \\ $$$$\Rightarrow \\ $$$${x}=\frac{\mathrm{36}}{\mathrm{25}}\wedge{y}=\frac{\mathrm{64}}{\mathrm{25}} \\ $$$${x}=\frac{\mathrm{9}}{\mathrm{400}}\wedge{y}=\frac{\mathrm{4}}{\mathrm{25}}\:\:\:\:\:_{\Rightarrow\:\mathrm{false}} ^{\left[\ast\right]\:{t}=\mathrm{4}} \\ $$$${x}=−\frac{\mathrm{7}}{\mathrm{16}}\pm\frac{\mathrm{3}}{\mathrm{2}}\mathrm{i}\wedge{y}=\frac{\mathrm{63}}{\mathrm{400}}\mp\frac{\mathrm{27}}{\mathrm{50}}\mathrm{i} \\ $$