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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 $Q(\vec{x})=2 x_{1}^{2}+2 x_{1} x_{2}+3 x_{2}-4$.
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