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# Minimizing a Path in the Coordinate plane

You start on point $$A=(0,0)$$ and you want to get to point $$B=(10,1)$$. There is a circular object with radius $$2$$ blocking your way: it's equation is $$(x-n)^2+y^2=4$$ for some $$n\in [2,8]$$. Let the shortest path from $$A$$ to $$B$$ such that you do not pass through the circular object have length $$P$$. What should $$n$$ be such that $$P$$ is minimized?

Maybe surprisingly, the answer is not $$n=8$$.

You can use Wolfram Alpha to bash it.

Note by Daniel Liu
3 years, 11 months ago

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For n = 2, distance = 8 + pi, which I guess is the shortest distance.

- 3 years, 11 months ago

Not quite! Good try.

Since nobody has replied, I will give the answer: the shortest distance is approximately $$10.4583$$ at $$n\approx 6.56261$$.

- 3 years, 11 months ago

After working out for nearly 4 hours, I found a smaller value. The answer is 10.3597 at n = 6.87425. I will post the solution tomorrow. The answer will be a bit smaller than this answer because of calculator limit I had to approximate.

- 3 years, 11 months ago