$\displaystyle A=\frac{1}{2}zy$ where $\displaystyle z=ky, k\geq0$

Thus,

$\displaystyle A(k, y)=\frac{1}{2}ky^2$

and

$\displaystyle A_1=\frac{1}{2}y^2$

$\displaystyle A_2=ky$

Not entirely sure how you got k=1 from these partials. Is that not how you determine the max of the area?