Maths Olympiad Prep

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Geometry Difficulty 5.2 AIME, harder Find the answer

Let A10A_{10} denote the answer to problem 10. Two circles lie in the plane; denote the lengths of the internal and external tangents between these two circles by xx and yy, respectively. Given that the product of the radii of these two circles is 15/215 / 2, and that the distance between their centers is A10A_{10}, determine y2−x2y^{2}-x^{2}.

A number or a short expression. Spacing and $ signs are ignored.

Solution

Suppose the circles have radii r1r_{1} and r2r_{2}. Then using the tangents to build right triangles, we have x2+(r1+r2)2=A102=y2+(r1−r2)2x^{2}+\left(r_{1}+r_{2}\right)^{2}=A_{10}^{2}=y^{2}+\left(r_{1}-r_{2}\right)^{2}. Thus, y2−x2=(r1+r2)2−(r1−r2)2=y^{2}-x^{2}=\left(r_{1}+r_{2}\right)^{2}-\left(r_{1}-r_{2}\right)^{2}= 4r1r2=304 r_{1} r_{2}=30

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