= Solution
<Fermat's principle> makes the <optical path length>, or transit time in a stationary medium, stationary under small variations of the <light ray> with fixed endpoints. It need not be a global minimum. In the result of part (a), the coefficient of $d^2$ is
$$
\frac1{2a}+\frac1{2b}-\alpha(n-1).
$$
For a family of paraxial <light rays> from the object to its image, stationarity with respect to $d$ requires this coefficient to vanish. Therefore the <thin lens> equation is
$$
\frac1a+\frac1b=2\alpha(n-1)=\frac1f.
$$
For a distant object, $a\to\infty$, and the image position $b=f$ defines the <focal length>. Hence
$$
\boxed{f=\frac1{2\alpha(n-1)}.}
$$
The axial member $d=0$ is stationary by symmetry for arbitrary $b$; requiring the neighbouring non-axial <light rays> to meet the same image is what determines $b$.
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