Use Lagrange multiply to find the local extremes of $f$

Let $f(x,y)=y^2-4xy+4x^2$ and $x^2+y^2=1$

**My work:**
Let $f:\mathbb{R}^2\rightarrow\mathbb{R}$ defined by $f(x,y)=y^2-4xy+4x^2$ and $g:\mathbb{R}^2\rightarrow\mathbb{R}$ defined by $g(x,y)=x^2+y^2-1$

We need define the function of Lagrange
$$L(x,y,\alpha)=f(x,y)+\alpha(g(x,y))$$

Then, calculating the critic point of $L$ we have

$\begin{equation}
-4y+8x+2x\alpha=0\\
2y-4x+2\alpha y=0\\
x^2+y^2-1=0
\end{equation}$

Then, solving the system, we have $x=0,\, \, y=0$

But i have a problem with this. this answer dont have sense because if $x=0$ and $y=0$ then $-1=0$

Can someone help me?