AMC 10 · 2020 · #7

Grade 11 geometry-2d
Archetype Coordinate Plane Figure Computation └ Sub-archetype Figure Constraint Equations for Unknown Quantity
slope-intercepttangent-addition-formulaquadratic-equationsoptimization convert-to-algebracaseworkextreme-principle ↑ Prerequisites: slope-intercepttrigonometric-ratios
📏 Medium solution 💡 3 insights
Problem
Two lines in the coordinate plane, neither horizontal nor vertical, cross each other at a 45 degree angle. One line's slope is exactly six times the other's. Among all pairs fitting this description, find the largest possible value of the product of the two slopes.

Pick an answer.

(A)
$\frac16$
(B)
$\frac23$
(C)
$\frac32$
(D)
3
(E)
6
How to solve
Strategy Convert to Algebra

The words "45^° angle" and "6 times the slope" are both statements about slopes, so if we name one slope m the whole picture becomes a single equation in m. The tangent formula for the angle between two lines is the bridge that turns the geometric 45^° into algebra. Once the equation is a quadratic, its few roots give a short list of candidate products, and picking the largest one finishes the problem.

1STEP 1

Name the two slopes

Write both with one letter.

m₁ = m, m₂ = 6m, m ≠ 0
2STEP 2

Write the angle between them

Use the tangent formula for the angle.

tan(θ₂ - θ₁) = (tanθ₂ - tanθ₁)/(1 + tanθ₁tanθ₂) = (6m - m)/(1 + 6m²) = 5m/(1 + 6m²)
3STEP 3

Turn 45 degrees into an equation

The absolute value makes two quadratics.

|5m/(1 + 6m²)| = tan 45^° = 1 ⟹ |5m| = 1 + 6m² ⟹ 6m² - 5m + 1 = 0 or 6m² + 5m + 1 = 0
4STEP 4

Solve both quadratics

Four slope candidates come out.

6m² - 5m + 1 = (2m - 1)(3m - 1) = 0 → m = 1/2 or m = 1/3; 6m² + 5m + 1 = (2m + 1)(3m + 1) = 0 → m = -1/2 or m = -1/3
5STEP 5

Compare the products

The largest product is three halves.

P = m · 6m = 6m²; m = ±1/2 → P = 3/2, m = ±1/3 → P = 2/3; P_max = 3/2
Answer
3/2
Test the winning pair directly: m = 1/2 gives slopes 1/2 and 3, and the second is indeed 6 times the first. The angle check gives |(3 - 1/2)/(1 + 3·1/2)| = |5/2/5/2| = 1, so the lines really do meet at 45^°, and their product is 1/2 · 3 = 3/2. The runner-up pair 1/3 and 2 also checks out with product 2/3, which is smaller, so 3/2 is the maximum and not merely one valid value.
💡Key takeaway

A slope is the tangent of a line's tilt, so an angle condition between two lines becomes an equation in their slopes, and once it is a quadratic you just check its few roots.

  • Name the two slopes
  • Write the angle between the lines
  • Turn 45^° into an equation
  • Solve both quadratics
  • Compare the products and pick the largest

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