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### Product of the radii

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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!
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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

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Associated Topics:
High School Conic Sections/Circles
High School Geometry

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