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#
