\($\frac{x}{y} = \frac{225}{xy} + \frac{y}{x}$\)
Rearrange as
x / y - y / x = 225 / xy
[ x^2 - y^2 ] / xy = 225 / xy
x^2 - y^2 = 225
( x + y) (x - y) = 225
The factors of 225 are : 1 | 3 | 5 | 9 | 15 | 25 | 45 | 75 | 225 (9 divisors)
So......the possible pair combos are
(x +y) (x - y) = 225
225 1
75 3
45 5
25 9
15 15
This appears to set up the following systems of equations :
x + y = 225
x - y = 1 add these and 2x = 226 → x = 113 and y = 112
x + y = 75
x + y = 3 add these and 2x = 78 → x = 39 and y = 36
x + y = 45
x - y = 5 add these and 2x = 50 → x = 25 and y = 20
x + y = 25
x - y = 9 add these and 2x = 34 → x = 17 and y = 8
x + y = 15
x - y = 15 add these and 2x = 30 → x= 15 and y = 0
So for x, y > 0......the pairs are
(113, 112) (39, 36) ( 25, 20) and (17, 8 )