AMC 10 · 2014 · #17

Grade 9 algebra
Archetype Arithmetic Expression Decomposition └ Sub-archetype Symmetric Function of Roots via Vieta
quadratic-equationsvieta-formulascoordinate-geometryslope-intercept convert-to-algebraextreme-principle ↑ Prerequisites: quadratic-equationscoordinate-geometry
📏 Long solution 💡 4 insights
Problem
Lines pivoting about a fixed point miss a curve for slopes in one interval. Find the sum of the edge slopes.

Pick an answer.

(A)
1
(B)
26
(C)
40
(D)
52
(E)
80
How to solve
Strategy Convert to Algebra

"The line does not touch the curve" is a geometric statement with nothing to compute. Substituting the line into the parabola converts it into a single quadratic equation in x, and then "does not touch" becomes "has no real root", which is a sign condition on the discriminant. That discriminant turns out to be a quadratic in m — the same m the question is about — so the whole problem collapses onto one inequality, Δ(m) ＜ 0. From there two things must be done, and only one of them is obvious. The obvious one is finding the endpoints. The less obvious one is proving the solution set really is a single nonempty open interval, because the problem's if-and-only-if phrasing quietly assumes it. Completing the square in m settles both at once. Finally, since only the sum of the endpoints is wanted, the symmetry of the completed square (or equivalently Vieta) gives it directly and the radical is never evaluated.

1STEP 1

Turn the line into an equation

The line and curve meet where a quadratic vanishes.

y = m(x - 20) + 14 ⟹ x² = mx - 20m + 14 ⟹ x² - mx + (20m - 14) = 0
2STEP 2

No crossing means no real root

Missing means the discriminant is negative.

Δ(m) = (-m)² - 4(20m - 14) = m² - 80m + 56, line misses P ⇔ Δ(m) ＜ 0
3STEP 3

Complete the square in m

That discriminant is itself a quadratic in the slope.

Δ(m) = (m - 40)² - 1544, 1544 = 4(20² - 14) ＞ 0, Δ(m) ＜ 0 ⇔ |m - 40| ＜ √(1544)
4STEP 4

Read the sum off the centre

Its roots sum straight off the coefficients.

r + s = 2 · 40 = 80; m² - 80m + 56 = (m - r)(m - s) = m² - (r + s)m + rs ⟹ r + s = 80; Q = (a, b): Δ(m) = m² - 4am + 4b, r + s = 4a
5STEP 5

Test real slopes

That sum is 80, choice (B).

m = 0: x² - 14 = 0 (meets); m = 1: x² - x + 6 = 0, Δ = -23 (misses); m = 80: x² - 80x + 1586 = 0, Δ = 56 (meets); r + s ≈ 0.706 + 79.294 = 80
Answer
80
The endpoints are 40 ∓ 2√(386) ≈ 0.7062 and 79.2938, and they add to 80 to full precision. A second, independent check exploits the derived fact that the y-coordinate of Q cannot affect the sum: move Q to (20, 0), still strictly below the parabola. The intersection equation becomes x² - mx + 20m = 0 with discriminant m² - 80m, whose roots are 0 and 80 — sum 80 again — and the midpoint slope m = 40 gives x² - 40x + 800 = 0 with discriminant -1600 ＜ 0, so that line genuinely misses. Two different points, same sum 4 · 20. The distractors are the predictable slips. Choice (C) 40 is the sum of the two tangency x-coordinates, 20 - √(386) and 20 + √(386), and also the centre of the interval; it is the answer for someone who stops one step early or forgets that the slope at (t, t²) is 2t rather than t. Choices (B) 26 and (D) 52 are 40 - 14 and its double, the kind of value that appears when the 20 and the 14 get combined; the general formula r + s = 4a shows the 14 cannot enter the sum at all. Choice (A) 1 is the coefficient of m², a Vieta misfire. One warning worth stating plainly: it is tempting to justify the interval by claiming something about the sign of m, but both endpoints here are positive, so no sign claim about m is doing any work. What does the work is the completed square, which proves the solution set is a nonempty interval, plus Vieta, which supplies its centre.
💡Key takeaway

"The line misses the curve" is the same sentence as "this quadratic has no real root", so chase the discriminant — and once it becomes a quadratic in the slope, its symmetry hands you the sum of the two edge slopes without ever finding them.

  • Turn the line into an equation
  • No crossing means no real root
  • Complete the square in m
  • Read the sum off the centre
  • Test real slopes

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