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Product of Radii of Two Circles


Date: 7/22/96 at 5:40:29
From: Matthew Darwent
Subject: Product of Radii of Two Circles

I have given up on solving this problem:

The length of a common internal tangent to two circles is 7, and a 
common external tangent is 11. Compute the product of the radii of the 
two circles.

Thanks,
Matthew


Date: 7/22/96 at 8:12:56
From: Doctor Anthony
Subject: Re: Product of Radii of Two Circles

Draw the diagram I describe and follow the calculation as set out 
below.

Let O1, O2 be the two centers distance 'd' apart, and let r1, r2 be 
the radii of the circles with r2>r1.

Join O1, O2 and draw the radii from the two centers to the points of 
contact with the common tangent of length 11 units.  From O1 draw a 
perpendicular to the radius r2 (just drawn) to meet this radius at 
point S.  Now we have a right angled triangle O1 S O2, with the 
shorter sides of length 11 and (r2-r1) and the hypotenuse equal to 
'd'.  Then by Pythagoras:

      d^2 = 11^2 + (r2-r1)^2  ......(1)

We now turn to the transverse tangent of length 7 units. Draw the 
lines from the centers O1, O2 to meet this tangent.  Now from the foot 
of the r2 radius where it meets the tangent draw a line parallel to 
'd' (line joining centers) until it meets the line of r1. This will 
require you to extend r1 backwards to the other side of O1.  Let T be 
the point where the lines meet. 

We now have another right angled triangle, with vertices at T and the 
two points of contact of the tangent with the circles. The shorter 
sides are of length 7 and (r2+r1), and the hypotenuse is 'd'.  
Again applying Pythagoras we get:

     d^2 = 7^2 + (r2+r1)^2   .......(2)

Now subtract (2) from (1) and we have:

      0 = 11^2 - 7^2 + (r2-r1)^2 - (r2+r1)^2

      0 = 121 - 49 + r2^2 - 2r2.r1 +r1^2 - r2^2 - 2r2.r1 - r1^2

      0 = 72 - 4r2.r1

 4r2.r1 = 72

  r2.r1 = 18

And so the product of the radii is 18.              

-Doctor Anthony,  The Math Forum
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Associated Topics:
High School Conic Sections/Circles
High School Geometry

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