Solving a Difficult Math Problem with Two Circles and Their Locus

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The problem involves finding the locus of midpoints of segments connecting points on two circles with radii 1 and 3, centered 10 units apart. The solution indicates that this locus is a line. Specifically, it is a line perpendicular to the line joining the centers of the circles. This line is located 3 units away from the intersection point on the circumference of either circle. The discussion emphasizes the geometric relationship between the circles and the midpoints of the segments connecting them.
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Two circles have radii 1 and 3 and centers a distance 10 apart. Find the locus of all points which are the midpoint of a segment with one end on each circle.





help ol' tongos solve this/
thanks
 
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It would be the line perpendicular to the line joining the two centres at a distance of 3 units from the point where the line intersects the circumference of either circle.
 
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