The joint distribution of X and Y is given by $f(x,y)=\frac{\exp(−y)}{y}$ where $0
So $f_Y(y) = \int_0^y \frac{\exp(−y)}{y} \mathrm{d}x = \exp(-y)$
which makes
$f_{X|Y}(x,y) = \frac{f(x,y)}{f_Y(y)}= \frac{\exp(-y)/y}{\exp(-y)} = \frac{1}{y}$
so
$\mathbb{E}(X^2+Y^2 |Y =y) = \int_{x=0}^y (x^2+y^2)\frac{f(x,y)}{f_Y(y)} \mathrm{d}x = \int_{x=0}^y \frac{(x^2+y^2)}{y}\mathrm{d}x = 2y^2$
Is this correct? I'm a bit confused?!