Product of the radii

Date: Wed, 3 Jul 96 20:09:00 NZST
From: Anonymous
Organization: Southern Lights BBS - Christchurch, NZ. 
Subject: help!

Hello!

How do you go about solving this problem? I'm stuck!

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!

Date: Wed, 3 Jul 1996 10:10:34 -0400 (EDT) 
From: Dr. Anthony
Subject: Re: help!

Draw the diagram I describe, and follow the argument below. 

Draw two circles, one of radius a, the other of radius b (b<a)with 
their centres a distance d apart. Draw in the direct common tangent 
(what you call external tangent), and also the radii from the centres 
to the points of contact with the tangent. From the centre of the 'b' 
circle draw a line parallel to the common tangent to meet the radius 
of the other circle. We now have a rectangle formed by this new 
line, the common tangent and the radii of the two circles (part of 
the radius in the case of the 'a' circle) 

The line we drew parallel to the common tangent has length 11 
units. This line, together with the line joining the centres and a 
length (a-b) cut off the radius of the 'a' circle, form a right angled 
triangle to which we can apply Pythagoras's theorem.

We have 11^2 = d^2 - (a-b)^2 = d^2 - (a^2 - 2ab + b^2) 

121 = d^2 + 2ab - a^2 - b^2 ..(1)

We now go through a similar exercise with the transverse common 
tangent (what you call the interior tangent), and we shall get the 
triangle with 

7^2 = d^2 - (a+b)^2 (The line from centre of 'b' circle, parallel to 
the transverse tangent, meets the 'a' radius outside that circle and 
cuts off a length (a+b) on that radius.)

This gives 49 = d^2 - 2ab - a^2 - b^2 ..(2) 

Now subtract equation (2) from (1) to get 

72 = 4ab and so ab = 18

So the product of the radii is 18 units.

-Doctor Anthony, The Math Forum