= Solution
Minimize or maximize $D(x)=x^Tx$ subject to $x^TAx=1$. A <Lagrange multiplier> $\Lambda$ gives
$$
2x-2\Lambda Ax=0.
$$
The constraint excludes $x=0$ and implies $Ax\neq0$, so it is a regular constraint surface. Multiplying the stationarity equation by $x^T$ gives $D=\Lambda>0$. Thus the <distance extrema on a real quadratic surface> satisfy
$$
\boxed{Ax=\frac1D x.}
$$
Every constrained extremum lies in a positive-eigenvalue <eigenspace>, with squared distance $D=1/\lambda$. If $A$ is positive definite, the global minimum is $1/\lambda_{\max}$ and the global maximum is $1/\lambda_{\min}$. For an indefinite or singular form the surface need not be bounded; <eigenvector> directions give stationary candidates, which may be saddles and need not supply both global extrema. With a repeated positive <eigenvalue>, there can be an entire family of equally distant stationary points.
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