= Solution
The azimuthal vector $\mathbf h_\phi$ is automatically orthogonal to the other two. Their remaining <dot product> is
$$
\mathbf h_\xi\cdot\mathbf h_\theta=[\sinh\xi\cosh\xi-g(\xi)g'(\xi)]\sin\theta\cos\theta.
$$
Thus <orthogonal coordinates> require $gg'=\sinh\xi\cosh\xi$, which integrates to $g^2=\sinh^2\xi+C$. The initial value gives $C=1$, and the smooth branch with $g(0)=1$ is
$$
\boxed{g(\xi)=\cosh\xi.}
$$
The corresponding coordinate-vector <norms> are $|\mathbf h_\xi|=|\mathbf h_\theta|=\sqrt{\sinh^2\xi+\sin^2\theta}$ and $|\mathbf h_\phi|=\sinh\xi\sin\theta$, as in the <hyperbolic parametrization of prolate spheroidal coordinates>. Coordinate degeneracies on the axis and focal segment are understood.
Eliminating the angular coordinates on the requested surface gives
$$
\boxed{\frac{x^2+y^2}{\sinh^21}+\frac{z^2}{\cosh^21}=1.}
$$
It is a <prolate spheroid> with long axis $z$, equatorial radius $\sinh1$, polar semiaxis $\cosh1$, and foci $(0,0,\pm1)$.
\Image[/past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2023/paper-1-prolate-spheroid.png]
{title=The <prolate spheroid> $\xi=1$ and its meridian <ellipse>, with equatorial and polar semiaxes and the two foci}
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