MathDB

10

Part of 2023 AMC 12/AHSME

Problems(2)

More Straightforward Algebra

Source: 2023 AMC12A #10

11/9/2023
Positive real numbers xx and yy satisfy y3=x2y^3 = x^2 and (y−x)2=4y2(y-x)^2 = 4y^2. What is x+yx+y?
<spanclass=′latex−bold′>(A)</span> 12<spanclass=′latex−bold′>(B)</span> 18<spanclass=′latex−bold′>(C)</span> 24<spanclass=′latex−bold′>(D)</span> 36<spanclass=′latex−bold′>(E)</span> 42<span class='latex-bold'>(A)</span>\ 12 \qquad <span class='latex-bold'>(B)</span>\ 18 \qquad <span class='latex-bold'>(C)</span>\ 24 \qquad <span class='latex-bold'>(D)</span>\ 36 \qquad <span class='latex-bold'>(E)</span>\ 42
AMCAMC 122023 AMC2023 AMC 12A
Slope of Radical Axis

Source: 2023 AMC 12B #10

11/15/2023
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}
analytic geometrygraphing linesslopeAMCAMC 12AMC 12 Bgeometry