Let \(\vec{x}\) and \(\vec{p}\) be points in \(\mathbb{R}^{n}\). Under what conditions on the scalar \(c\) is the set \[ \|\vec{x}\|^{2}+2\langle\vec{p}, \vec{x}\rangle+c=0 \]

Question

a) Let \(\vec{x}\) and \(\vec{p}\) be points in \(\mathbb{R}^{n}\). Under what conditions on the scalar \(c\) is the set
\[
\|\vec{x}\|^{2}+2\langle\vec{p}, \vec{x}\rangle+c=0
\]
a sphere \(\left\|\vec{x}-\vec{x}_{0}\right\|=R \geq 0\) ? Compute the center, \(\vec{x}_{0}\), and radius, \(R\), in terms of \(\vec{p}\) and \(c\).
b) Let
\[
\begin{aligned}
Q(\vec{x}) &=\sum a_{i j} x_{i} x_{j}+2 \sum b_{i} x_{i}+c \\
&=\langle\vec{x}, A \vec{x}\rangle+2\langle\vec{b}, \vec{x}\rangle+c
\end{aligned}
\]
be a real quadratic polynomial so \(\vec{x}=\left(x_{1}, \ldots, x_{n}\right), \vec{b}=\left(b_{1}, \ldots, b_{n}\right)\) are real vectors and \(A=\left(a_{i j}\right)\) is a real symmetric \(n \times n\) matrix. Just as in the case \(n=1\) (which you should do first), if \(A\) is invertible find a vector \(\vec{v}\) (depending on \(A\) and \(\vec{b}\) ) so that the change of variables \(\vec{y}==\vec{x}-\vec{v}\) (this is a translation by the vector \(\vec{v}\) ) so that in the new \(\vec{y}\) variables \(Q\) has the simpler form
\[
Q=\langle\vec{y}, A \vec{y}\rangle+\gamma \text { that is, } Q=\sum a_{i j} y_{i} y_{j}+\gamma,
\]
where \(\gamma=c-\left\langle\vec{b}, A^{-1} \vec{b}\right\rangle\).
As an example, apply this to \(Q(\vec{x})=2 x_{1}^{2}+2 x_{1} x_{2}+3 x_{2}-4\).