AMC 12A 2025 (Problem 9)Let www be the complex number 2+i2+i2+i, where i=−1i=\sqrt{-1}i=−1. What real number rrr has the property that rrr, www, and w2w^2w2 are three collinear points in the complex plane?(A) 34\text{(A)}\;\frac{3}{4}(A)43(B) 1\text{(B)}\;1(B)1(C) 75\text{(C)}\;\frac{7}{5}(C)57(D) 32\text{(D)}\;\frac{3}{2}(D)23(E) 53\text{(E)}\;\frac{5}{3}(E)35Related TopicsCoreToolkit 97 — Basic Complex NumbersMajorToolkit 95 — Coordinate GeometryHints (5)Hint 1w=2+iw=2+iw=2+iw2=(2+i)2=4+4i+i2=4+4i−1=3+4iw^2=(2+i)^2=4+4i+i^2=4+4i-1=3+4iw2=(2+i)2=4+4i+i2=4+4i−1=3+4iHint 2Hint 3Let ddd be the line passing through w(2,1)w(2,1)w(2,1) and w2(3,4)w^2(3,4)w2(3,4). Find the equation of ddd.Hint 4By Toolkit 95 — Coordinate Geometry:sloped=4−13−2=3\operatorname{slope}_d=\frac{4-1}{3-2}=3sloped=3−24−1=3d:y−1=3(x−2)⇒y=3x−5d:\quad y-1=3(x-2)\Rightarrow y=3x-5d:y−1=3(x−2)⇒y=3x−5Hint 5Find the xxx-intercept:y=0⇒0=3r−5⇒r=53y=0\Rightarrow0=3r-5\Rightarrow r=\frac{5}{3}y=0⇒0=3r−5⇒r=35Final Answer(E) 53\frac{5}{3}35Related Problems (4)AMC 12A 2025 (Problem 17)AMC 10A 2025 (Problem 20)AMC 10A 2025 (Problem 22)AMC 12A 2025 (Problem 11)
