Did I solve this Lagrange multiplier correctly? (Minimizing $f(x,y)=x^2+y^2$ with constraint $x+2y-5=0$ and $x>0$ and $y>0$.)

Question

Okay, so we're given this equation $f(x,y)=x^2+y^2$ and the constraint is $x+2y-5=0$, and that $x>0$ and $y>0$,we have to find the minimum.

My answer is $(1,2)$, and my work is shown below.

Can anyone just check if I'm doing this correctly?