Find max and min subject to constraint ||x|| = 4

Question

$Q(x,y)=7x^{2}+12xy+12y^{2}$

I only know how to do this is $\|(x,y)\|=1$

If $\|(x,y)\|=1$, the eigenvalues are $16$ and $3$. So obviously $\min=3,\max=16$.

I don't know what to do if $\|(x,y)\|=4$, and cannot find any examples.

$\|(x,y)\|=4$, it means that $x^{2}+y^{2}=4$. How does that affect my calculations?

Answer 1

If $\|(x_0,y_0)\| = 1$, then $\|(4x_0,4y_0)\| = 4$ and $Q(4x_0,4y_0) = 16Q(x_0,y_0)$.

Thus, if $(x_0,y_0)$ maximizes/minimizes $Q(x,y)$ with the constraint $\|(x,y)\| = 1$, then $(4x_0,4y_0)$ maximizes/minimizes $Q(x,y)$ with the constraint $\|(x,y)\| = 4$.

Also, the maximium/minimum value of $Q(x,y)$ with the constraint $\|(x,y)\| = 4$ will be $16$ times the maximium/minimum value of $Q(x,y)$ with the constraint $\|(x,y)\| = 1$.