I think this is the wrong interpretation; your answer would be correct if the rope were hovering at a height of 10 km along the full length.

My interpretation is that the rope is pulled taut to a height of 10 km at a single point, so that it runs in a straight line to the horizon in either direction and lays on the ground the rest of the way; this is also relevant to the question of longest line of sight from a given height.

Actually, it's a different question. Check out the second diagram linked to below: https://news.ycombinator.com/item?id=14863949

So yes, you could supply the answer in terms of r radius, but you do need to know the radius unless I'm missing something obvious.

That diagram wasn't posted by the GP.

Yes, but I only posted that so you'd see which problem the GP is referring to: the second one. Unless somehow the rope is some special kind of non-flexible rope that always assumes the shape of a circle.

What makes you think that's the problem the GP was referring to? They haven't posted at all again in this thread to confirm exactly what was meant.