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Common Tangent

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March 27, 2008
geometrysimilar triangles

Problem Statement

Let C1 C_1 and C2 C_2 be circles defined by (x \minus{} 10)^2 \plus{} y^2 \equal{} 36and (x \plus{} 15)^2 \plus{} y^2 \equal{} 81,respectively. What is the length of the shortest line segment PQ‾ \overline{PQ} that is tangent to C1 C_1 at P P and to C2 C_2 at Q Q? <spanclass=′latex−bold′>(A)</span> 15<spanclass=′latex−bold′>(B)</span> 18<spanclass=′latex−bold′>(C)</span> 20<spanclass=′latex−bold′>(D)</span> 21<spanclass=′latex−bold′>(E)</span> 24 <span class='latex-bold'>(A)</span>\ 15 \qquad <span class='latex-bold'>(B)</span>\ 18 \qquad <span class='latex-bold'>(C)</span>\ 20 \qquad <span class='latex-bold'>(D)</span>\ 21 \qquad <span class='latex-bold'>(E)</span>\ 24
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