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2. Consider a medium in which the refractive index $n$ is inversely proportional to $\rho$, that is $n = n_0/\rho$, where $\rho$ is the distance from the origin. Use Fermat's principle, that the integral \begin{equation} \int_1^2 n(x, y)ds = \int_1^2 n(x,y)\sqrt{dx^2 + dy^2} \end{equation} is stationary, to find the path of a light ray travelling in a plane containing the origin. [Hint: Use two-dimensional polar coordinates and write the path as $\phi = \phi(\rho)$. The Fermat integral should have the form $\int f(\phi, \phi', \rho) d\rho$, where $f(\phi, \phi', \rho)$ is actually independent of $\phi$.]

          2. Consider a medium in which the refractive index $n$ is inversely proportional to $\rho$, that is $n = n_0/\rho$, where $\rho$ is the distance from the origin. Use Fermat's principle, that the integral \begin{equation*} \int_1^2 n(x, y)ds = \int_1^2 n(x,y)\sqrt{dx^2 + dy^2} \end{equation*} is stationary, to find the path of a light ray travelling in a plane containing the origin. [Hint: Use two-dimensional polar coordinates and write the path as $\phi = \phi(\rho)$. The Fermat integral should have the form $\int f(\phi, \phi', \rho) d\rho$, where $f(\phi, \phi', \rho)$ is actually independent of $\phi$.]

$2. Consider a medium in which the refractive index n is inversely proportional to ρ, that is n = n0/ρ, where ρ is the distance from the origin. Use Fermat's principle, that the integral ∫1^2 n(x, y)ds = ∫1^2 n(x,y)√(dx^2 + dy^2) is stationary, to find the path of a light ray travelling in a plane containing the origin. [Hint: Use two-dimensional polar coordinates and write the path as ϕ = ϕ(ρ). The Fermat integral should have the form ∫ f(ϕ, ϕ', ρ) dρ, where f(ϕ, ϕ', ρ) is actually independent of ϕ.]$

Added by James M.

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