The inner circle

Geometry Level 3

A circle of radius $a$ is drawn tangent to the $x$ and $y$ axes. This is circle #1, shown above as the largest, red circle.

Circle #2 is a smaller circle drawn tangent to the $x$ and $y$ axes and to circle #1.

Circle #3 is an even smaller circle tangent to the $x$ and $y$ axes, and to circle #2.

In general, circle #N is drawn tangent to the $x$ and $y$ axes and to circle #(N-1).

Which decimal is closest to the ratio of the area of circle #101 to the area of circle #100?

0.01 0.04 0.03 0.02 0.015 0.035 0.045 0.025

1 solution

Brian Kardon
Feb 15, 2015

First, find the radius of circle #2 in terms of a. The distance from the origin to the point of tangency between circle #2 and circle #1 is

$d = (\sqrt{2} - 1)a$ .

This distance is related to the radius of circle #2 by

$d = r + \sqrt{2} r$ ,

giving us that

$r = \frac{d}{\sqrt{2} + 1}$ .

In turn, that gives us a relationship between $r$ and $a$ :

$r = k a$

where

$k = \frac{\sqrt{2} - 1}{\sqrt{2}+1}$ .

By inspection, the radii of circles #3 and #2 should have the same ratio of $k$ between them; thus the ratio of the radii of circles #(N+1) and #N should be $k$ , and the ratio of the areas of circles #(N+1) and #N should be $k^2$ . This is a general formula, so it should hold for circles #101 and #100.

This gives us a value of approximately $0.029$ , which is closest to $0.03$ .

The inner circle

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