Question

solution of under root x / Y + under root Y / x = 13 / 6 and xY = 36

Answer 1

x y = 36    => y = 36/x.

So LHS =  √(x²/36) + √(36/x²) = √(13/6) = RHS
              6/x + x/6 = √(13/6)
      So      x² + 36 = x √78
                x² - √78 x + 36 = 0
               x = [√78 +- √(78 - 144) ] /2
                   = √78 /2 + √66 /2 i 
There does not seem to be a real number solution to the given equation...

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If the RHS is   13/6 instead of  √(13/6) then there is a solution..

        x/6 + 6/x = 13/6
        x² - 13 x + 36 = 0
        x = 9 or 4...   Then  y = 4 or 9...