I am working on finding the largest rectangle inscribed in $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$. Then its given (this is an example, so answer given)
Maximize $A(x,y) = 4xy$.
$f_x=\lambda g_x \rightarrow 4y = \frac{2\lambda x}{a^2}$
$f_y = \lambda g_y \rightarrow 4x = \frac{2 \lambda y}{b^2}$
$\frac{y}{x} = \frac{xb^2}{ya^2} \rightarrow \frac{x}{y} = \frac{a}{b}$
Then question is why in the next step:
$x=ak,\qquad y=bk?$
Where did $a, b$ come from? Why isit not $x=a, y=b$?