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Question

Three circles of radius s are drawn in the first quadrant of the xy-plane. The first circle is tangent to both axes, the second is tangent to the first circle and the x-axis, and the third is tangent to the first circle and the y-axis. A circle of radius r>s is tangent to both axes and to the second and third circles. What is r/s? FIGURE

Accepted Answer

FIGURE Consider a right triangle as shown. Applying the Pythagorean theorem yields (r+s)^2=(r-3s)^2+(r-s)^2 Simplifying, align* r^2+2rs+s^2&=r^2-6rs+9s^2+r^2-2rs+s^2 0&=r^2-10rs+9s^2 &=(r-9s)(r-s) align*But we know that r s, so the only solution is r = 9s; hence r/s = 9.