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Question Number 89994 by jagoll last updated on 20/Apr/20

x−y=3(√(xy))  ((x/y)−1)^3 +((y/x)−1)^3  = ?

$$\mathrm{x}−\mathrm{y}=\mathrm{3}\sqrt{\mathrm{xy}} \\ $$$$\left(\frac{\mathrm{x}}{\mathrm{y}}−\mathrm{1}\right)^{\mathrm{3}} +\left(\frac{\mathrm{y}}{\mathrm{x}}−\mathrm{1}\right)^{\mathrm{3}} \:=\:? \\ $$

Answered by MJS last updated on 21/Apr/20

y=px  ((x/(px))−1)^3 +(((px)/x)−1)^3 =(((p−1)^4 (p^2 +p+1))/p^3 )  y+3(√(xy))−x=0  xp+3∣x∣(√p)−x=0  ⇒ p=((11±3(√(13)))/2)  ⇒ answer is 972

$${y}={px} \\ $$$$\left(\frac{{x}}{{px}}−\mathrm{1}\right)^{\mathrm{3}} +\left(\frac{{px}}{{x}}−\mathrm{1}\right)^{\mathrm{3}} =\frac{\left({p}−\mathrm{1}\right)^{\mathrm{4}} \left({p}^{\mathrm{2}} +{p}+\mathrm{1}\right)}{{p}^{\mathrm{3}} } \\ $$$${y}+\mathrm{3}\sqrt{{xy}}−{x}=\mathrm{0} \\ $$$${xp}+\mathrm{3}\mid{x}\mid\sqrt{{p}}−{x}=\mathrm{0} \\ $$$$\Rightarrow\:{p}=\frac{\mathrm{11}\pm\mathrm{3}\sqrt{\mathrm{13}}}{\mathrm{2}} \\ $$$$\Rightarrow\:\mathrm{answer}\:\mathrm{is}\:\mathrm{972} \\ $$

Answered by john santu last updated on 21/Apr/20

x^2 +y^2 −2xy = 9xy  x^2 +y^2  = 11xy   ((x^2 +y^2 )/(xy)) = 11 ⇒ (x/y)+(y/x) = 11  ((x/y)−1)^3 +((y/x)−1)^3 =   ((x/y)+(y/x)−2)((((x/y)+(y/x)−2))^2 −3((x/y)−1)((y/x)−1))  (9)(9^2 −3(2−((x/y)+(y/x)))=  (9)(81+27)= 9×108 = 972

$${x}^{\mathrm{2}} +{y}^{\mathrm{2}} −\mathrm{2}{xy}\:=\:\mathrm{9}{xy} \\ $$$${x}^{\mathrm{2}} +{y}^{\mathrm{2}} \:=\:\mathrm{11}{xy}\: \\ $$$$\frac{{x}^{\mathrm{2}} +{y}^{\mathrm{2}} }{{xy}}\:=\:\mathrm{11}\:\Rightarrow\:\frac{{x}}{{y}}+\frac{{y}}{{x}}\:=\:\mathrm{11} \\ $$$$\left(\frac{{x}}{{y}}−\mathrm{1}\right)^{\mathrm{3}} +\left(\frac{{y}}{{x}}−\mathrm{1}\right)^{\mathrm{3}} =\: \\ $$$$\left(\frac{{x}}{{y}}+\frac{{y}}{{x}}−\mathrm{2}\right)\left(\left(\left(\frac{{x}}{{y}}+\frac{{y}}{{x}}−\mathrm{2}\right)\right)^{\mathrm{2}} −\mathrm{3}\left(\frac{{x}}{{y}}−\mathrm{1}\right)\left(\frac{{y}}{{x}}−\mathrm{1}\right)\right) \\ $$$$\left(\mathrm{9}\right)\left(\mathrm{9}^{\mathrm{2}} −\mathrm{3}\left(\mathrm{2}−\left(\frac{{x}}{{y}}+\frac{{y}}{{x}}\right)\right)=\right. \\ $$$$\left(\mathrm{9}\right)\left(\mathrm{81}+\mathrm{27}\right)=\:\mathrm{9}×\mathrm{108}\:=\:\mathrm{972} \\ $$

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