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Given two circles, one of radius 1, and another of radius 2, and the distance between the centers is 4, what is set of points that are equidistant from the two circles?

Now I think naturally the first thing to do is to solve an easier problem, so if one matches the radii, then the set of points that are equidistant is just a straight line that is exactly half-way in between them. I'm thinking of what happens when we increase the radius of one of them, the line somehow rotates towards the circle with the smaller radius, and the line also becomes closer to the circle of the larger radius. Is my intuition correct here?

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Yes, your intuition is correct. If we put the large circle at the origin and the small one centered at $(4,0)$ we are looking for the soluion of $\sqrt{x^2+y^2}-2=\sqrt{(x-4)^2+y^2}-1$. It looks like this:when plotted in Alpha. The equation is $y=\pm \frac 12 \sqrt{15}\sqrt{4x^2-16x+15}$ also obtained from Alpha

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The line will "bend" towards the smaller circle and become a branch of a hyperbola with focal points at the centers of the two circles. You can immediately see this from one of the equivalent definitions of a hyperbola.

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