Question

If √x/y + √y/x = 10/3, then find xy.

Answer 1

Squaring on both sides we get

x/y +y/x +2 =100/9

Subtract 2 from both sides:
x/y + y/x = 82/9

Combine the fractions:
(x^2 + y^2) / xy = 82/9

From here by inspection, we can see that at least two solutions are (1, 9) and (9, 1) making xy = 9. But we will continue by factoring:

9x^2 + 9y^2 = 82xy
9x^2 - 82xy + 9y^2 = 0
9x^2 - 81xy - xy + 9y^2 = 0
9x(x - 9y) - y(x - 9y) = 0
(9x - y)(x - 9y) = 0
x = 9y OR y = 9x

There are an infinite amount of solutions for xy, as long as y = 9x or x = 9y such as (1, 9) = 9, (18, 2) = 36, (4, 36) = 144