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Positive real numbers \(x,y\) satisfy the equations \(x^2 + y^2 = 1\) and \(x^4 + y^4= \frac{17}{18}\). Find \(xy\).

 Dec 13, 2019
 #1
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xy = 5/2.

 Dec 13, 2019
 #2
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Square both sides of the first equation:

 

(x2+y2)2 = 12    or   x4+ 2(xy)2 + y4 = 1

 

So:  17/18 + 2(xy)2 = 1 

 

       2(xy)2 = 1/18

 

         (xy)2 = 1/36

 

         xy = 1/6   or   xy = -1/6

.

 Dec 13, 2019

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