What is the locus of points in a plane equidistant from a circle?

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I was working on an exam review packet for high school and I saw a question about locuses of points equidistant from a circle and lying on a given line. My confusion is that the answer key is inconsistent with the solutions the teacher put online. Also, I emailed the teacher, and I'm still waiting on his take. While I'm waiting, may I get a few suggestions?

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Without additional constraint the locus of points equidistant (at distance d) from a circle (of radius r) is the union of two circles of radius $r\pm d$.

So you have to distinguish for cases:

  • $d>r$ in this case there is only one circle.
  • $d=r$ there is one circle and a point.
  • $d<r$ there are two circles.

Now if you add the constraint that the points also belong to a line, then you have to look whether the line passes through the common center, is tangent or crosses the various circles involved, or is too far to have intersection points.

When you put all this together, you will find $0,1,2,3$ or $4$ possible points of intersection.