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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.

 
 Aug 1, 2022
 #1
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x^2 + y^2   = 1        square both sides

 

x^4 + 2x^2y^2  + y^2  =  1

 

(x^4 + y^4)  +  2(xy)^2    =1

 

(17/19)  + 2(xy)^2   =  1

 

2(xy)^2  =   1 -  17/19

 

2(xy)^2   = 2/19             divide through by 2

 

(xy)^2  =    1/19            take the positive root

 

xy  = sqrt (1/19)  =   sqrt (19) / 19

 

cool cool cool

 
 Aug 1, 2022

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