Let \(S\subset\RR^3\) denote the ellipsoidal surface defined by
\[ 2x^2+(y-1)^2+(z-10)^2=1. \]
Let \(T\subset\RR^3\) be the surface defined by
\[ z=\frac{1}{x^2+y^2+1}. \]
Prove that there exist points \(p\in S\), \(q\in T\), such that the line \(pq\) is perpendicular to \(S\) at \(p\) and to \(T\) at \(q\).
Solution
Let \[ D(p,q)=\norm{p-q}^2, \qquad (p,q)\in S\times T, \] and for a point \(r\) of a surface \(\Sigma\) write \(\operatorname{Tan}_r\Sigma\) for the tangent plane of \(\Sigma\) at \(r\).
1. The function \(D\) attains its minimum on \(S\times T\) at some pair\((p,q)\).
The ellipsoid \(S\) is compact, so there is \(R>0\) such that \[ S\subset B_R(0). \] The surface \(T\) is the graph of the continuous function \[ (x,y)\longmapsto \frac1{x^2+y^2+1}, \] so \(T\) is closed.
Fix any pair \((p_0,q_0)\in S\times T\) and put \(M=\norm{p_0-q_0}\). If \(q\in T\) satisfies \(\norm{q}>R+M+1\), then for every \(p\in S\), \[ \norm{p-q}\ge \norm{q}-\norm{p}>M+1>M. \] Hence the infimum of \(D\) on \(S\times T\) equals its infimum on \[ S\times\bigl(T\cap \overline B_{R+M+1}(0)\bigr), \] which is compact. By continuity, \(D\) attains its minimum there, at a pair \((p,q)\), and this pair minimizes \(D\) on \(S\times T\).
2. The points \(p\) and \(q\) are distinct.
Every point \((x,y,z)\in S\) satisfies \[ (z-10)^2\le1, \] so \[ 9\le z\le11. \] Every point \((x,y,z)\in T\) satisfies \[ 0<z=\frac1{x^2+y^2+1}\le1. \] Thus \(S\cap T=\varnothing\), so \(p\ne q\) and the line \(pq\) is defined.
3. The line \(pq\) is perpendicular to \(S\) at \(p\).
Let \(v\in\operatorname{Tan}_pS\). Choose a smooth curve \(\gamma:(-\varepsilon,\varepsilon)\to S\) with \[ \gamma(0)=p, \qquad \gamma'(0)=v. \] Because \((p,q)\) minimizes \(D\), the function \[ \varphi(t)=\norm{\gamma(t)-q}^2 \] has a minimum at \(t=0\). Hence \[ 0=\varphi'(0)=2(p-q)\cdot v. \] Since this holds for every \(v\in\operatorname{Tan}_pS\), the vector \(q-p\) is orthogonal to \(\operatorname{Tan}_pS\).
4. The line \(pq\) is perpendicular to \(T\) at \(q\).
Let \(w\in\operatorname{Tan}_qT\) and choose a smooth curve \(\eta:(-\varepsilon,\varepsilon)\to T\) with \[ \eta(0)=q, \qquad \eta'(0)=w. \] The function \[ \psi(t)=\norm{p-\eta(t)}^2 \] has a minimum at \(0\), so \[ 0=\psi'(0)=2(q-p)\cdot w. \] Hence \(q-p\) is orthogonal to \(\operatorname{Tan}_qT\).
5. Q.E.D.
Steps 1 and 2 give distinct points \(p\in S\) and \(q\in T\), and steps 3 and 4 show that the line \(pq\) is perpendicular to \(S\) at \(p\) and to \(T\) at \(q\).