AMC 10 · 2025 · #9

Grade 11 algebra

Archetype Coordinate Plane Figure Computation └ Sub-archetype Line Intersection Point as the Answer

complex-numberscoordinate-geometryslope-intercept convert-to-algebra ↑ Prerequisites: complex-numbers

📏 Short solution 💡 1 insight

Problem

The number w = 2 + i and its square w² are two fixed points in the complex plane. A real number r sits on the real axis. Find the value of r that makes the three points r, w, and w² lie on one straight line.

Pick an answer.

(A)

$\frac34$

(B)

(C)

$\frac75$

(D)

$\frac32$

(E)

$\frac53$

How to solve

Strategy Convert to Algebra

The word 'collinear' is geometric, but each complex number is just a point (a, b), so I can turn the geometry into algebra. First I fix the two known points w and w² as coordinates, write the equation of the line through them, and then read off where that line crosses the real axis. Picturing the points (Draw a Diagram) keeps the setup honest, and splitting the work into 'square w', 'find the line', 'find the crossing' (Identify Subproblems) keeps each step small.

1STEP 1

Square w

Square w = 2 + i, using i² = -1, to get the second known point w² = 3 + 4i.

w² = (2+i)² = 4 + 4i + i² = 4 + 4i - 1 = 3 + 4i

2STEP 2

Plot the two known points

Read a + bi as the point (a, b): w is (2, 1), w² is (3, 4), and real r is (r, 0) on the horizontal axis.

w ⇔ (2, 1), w² ⇔ (3, 4), r ⇔ (r, 0)

3STEP 3

Find the line through w and w²

Two points fix a line: from (2, 1) to (3, 4) the slope is 3, giving the line y = 3x - 5.

m = (4-1)/(3-2) = 3 → y - 1 = 3(x - 2) → y = 3x - 5

4STEP 4

Find where the line meets the real axis

Set y = 0 and solve for x: the line crosses the real axis at r = $\frac{5}{3}$ , which is choice (E).

0 = 3x - 5 → x = 5/3 → r = 5/3

Answer

5/3

Substitute back: at x =

\frac{5}{3}

, y = 3(

\frac{5}{3}

) - 5 = 5 - 5 = 0, so (

\frac{5}{3}

, 0) really is on the line y = 3x - 5. The three points (

\frac{5}{3}

, 0), (2, 1), and (3, 4) each rise by 3 for every step of 1 to the right, so all three share the slope 3 and are genuinely collinear. The value

\frac{5}{3}

is a bit less than 2, which fits: the line goes up-right, so it must cross the axis just to the left of w = (2, 1). This matches choice (E).

💡Key takeaway

Every complex number is just a dot on a grid, so 'collinear' means find the line through the two dots and see where it crosses the real axis.

Square w
Plot the two known points
Find the line through w and w²
Find where the line meets the real axis

Square w

Plot the two known points

Find the line through w and w²

Find where the line meets the real axis

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