Calling a = (x/y)^(1/2) in
(x/y)^(1/2) + (y/x)^(1/2) = 17/4
x/(y^(1/2)) + y/(x^(1/2)) = 65/4
we obtain
((a+1/a=17/4),(a sqrt(x)+sqrt(y)/a=65/4))
Solving for a,x we have
((a = 4, x = 1/256 (4225 - 130 sqrt[y] + y)),(a = 1/4, x = 4225 - 2080 sqrt[y] + 256 y))
The first solution gives
a = 4 so a^2 = x/y = 16 them follows
x=16y=(4225 - 130 sqrt[y] + y)/256 or
(16 xx 256 y - y -4225)^2=(-130sqrt(y))^2 or
3969 y^2- 8194 y +4225=0
with roots {color(red)(y = 1),color(red)(x=16)} and {y = 4225/3969, x = (16 xx4225)/3969 }
The second solution gives
a = 1/4 so a^2=x/y=1/16 following
x = y/16 =4225 - 2080 sqrt[y] + 256 y or
y = 16 (4225 - 2080 sqrt[y] + 256 y) or
(y-16 xx 256 y - 16 xx 4225)^2=2080^2y or
3969 y^2 - 131104 y + 1081600=0
with roots {color(red)(y = 16), color(red)(x = 1)}and {y = 67600/3969,x =67600/(3969 xx 16)}
Ressuming, the real feasible solutions are:
((x = 1, y = 16),(x = 16, y = 1))
Attached the solutions region.
In blue
x/(y^(1/2)) + y/(x^(1/2)) = 65/4
and in red
(x/y)^(1/2) + (y/x)^(1/2) = 17/4
![enter image source here]()
