Question

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. ]

Answer 1

(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 \(\left.a x^{2}+2 b x+c=0 .\right]\)