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Slope of Radical Axis

Source: 2023 AMC 12B #10

November 15, 2023
analytic geometrygraphing linesslopeAMCAMC 12AMC 12 Bgeometry

Problem Statement

In the xyxy-plane, a circle of radius 44 with center on the positive xx-axis is tangent to the yy-axis at the origin, and a circle with radius 1010 with center on the positive yy-axis is tangent to the xx-axis at the origin. What is the slope of the line passing through the two points at which these circles intersect?
<spanclass=′latex−bold′>(A)</span> 27<spanclass=′latex−bold′>(B)</span> 37<spanclass=′latex−bold′>(C)</span> 229<spanclass=′latex−bold′>(D)</span> 129<spanclass=′latex−bold′>(E)</span> 25<span class='latex-bold'>(A)</span>\ \dfrac{2}{7} \qquad<span class='latex-bold'>(B)</span>\ \dfrac{3}{7} \qquad<span class='latex-bold'>(C)</span>\ \dfrac{2}{\sqrt{29}} \qquad<span class='latex-bold'>(D)</span>\ \dfrac{1}{\sqrt{29}} \qquad<span class='latex-bold'>(E)</span>\ \dfrac{2}{5}
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