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. ]