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Let $A$ be a positive definite $n \times n$ real matrix, $\vec{b}$ a real vector, and $\vec{N}$ a real unit vector.
a) For which value(s) of the real scalar $c$ is the set
$E:=\left\{\vec{x} \in \mathbb{R}^{3} \mid\langle\vec{x}, A \vec{x}\rangle+2\langle\vec{b}, \vec{x}\rangle+c=0\right\}$
(an ellipsoid) non-empty?

b) For what value(s) of the scalar $d$ is the plane $Z:=\left\{\vec{x} \in \mathbb{R}^{3} \mid\langle\vec{N}, \vec{x}\rangle=d\right\}$ tangent to the above ellipsoid $E$ (assumed non-empty)?

[SUGGESTION: First discuss the case where $A=I$ and $\vec{b}=0$. Then show how by a change of variables, the general case can be reduced to this special case. ]
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(a) $c \leq\left\langle b, A^{-1} b\right\rangle$. If $n=1$, this of course reduces to a familiar condition.

(b) $d=-\left\langle\vec{N}, A^{-1} \vec{b}\right\rangle \pm \sqrt{\left\langle\vec{N}, A^{-1} \vec{N}\right\rangle} \sqrt{\left\langle\vec{b}, A^{-1} \vec{b}\right\rangle-c} .$
For $n=1$ this is just the solution $d=\frac{-b \pm \sqrt{b^{2}-a c}}{a}$ of the quadratic equation $\left.a x^{2}+2 b x+c=0 .\right]$
by Platinum (101k points)

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