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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.


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 

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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