If the parabolic mirror is the the perfect collector/reflector is the convex side a perfect diffuser?

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I was wondering about the scattering of light (or other rays) from the convex side of a parabolic mirror. In particular, whether the reflected rays appear to originate from the focus (of course I'm thinking of incoming rays parallel to the parabola's axis of symmetry). After some simple trigonometric considerations, I reckoned that that is indeed the case (see the figure below). In that case, the outward reflected light is the same as from an isotropic source and is as uniform as can be.

My question is whether this is a well known property of the parabola. A cursory search on the internet came up with zilch.

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