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The Midpoint Locus (Posted on 2019-10-11) Difficulty: 4 of 5
Let C1 and C2‚Äč be circles with centres 10 units apart, with radii of length 1 and 3 respectively. Firstly find the locus of all points, M, for which there exist points X on C1 and Y on C2 such that M is the midpoint of the line segment XY. Then find the area of the closed region inscribed by the locus.

No Solution Yet Submitted by Danish Ahmed Khan    
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Solution Solution | Comment 1 of 3
The locus is an annulus.  The 10 unit displacement of the centres is irrelevant.  

Let the larger circle have radius r2 and the smaller r1.  If the point on the smaller circle were fixed, the locus would be a circle of radius r2/2.  If the point on the larger circle were fixed, the locus would be a circle of radius r1/2.

The locus of M: the outer circle has radius (r2+r1)/2 and the inner circle has radius (r2-r1)/2.

Square and subtract these to get the area r1r2*pi

For this problem the area is 3pi

  Posted by Jer on 2019-10-16 12:39:46
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