AMC 10 · 2004 · #18

Grade 8 geometry-2dalgebra
Archetype Coordinate Plane Figure Computation └ Sub-archetype Pythagorean / Distance Formula for a Length
midpoint-formulaquadratic-equationspythagorean-theorem symmetry-argumentconvert-to-algebra ↑ Prerequisites: midpoint-formula
📏 Medium solution 💡 2 insights
Problem
Two points lie on the curve y equals 4x squared plus 7x minus 1. The origin sits exactly halfway between them. Find the distance between the two points.

Pick an answer.

(A)
$2\sqrt5$
(B)
$5+\frac{\sqrt2}{2}$
(C)
$5+\sqrt2$
(D)
7
(E)
$5\sqrt2$
How to solve
Strategy Introduce a Variable

Two unknown points on a curve looks like four unknown numbers, but the midpoint condition cuts that in half: if A=(p,q) then B has to be (-p,-q). So name A and write down the one demand left - that both A and its half-turn image lie on the curve. That is two equations in p and q, and instead of solving them one at a time it pays to add them and subtract them. Adding kills the odd-degree terms and subtracting kills the even-degree ones, so each combination isolates one unknown.

1STEP 1

Let the midpoint fix B from A

The midpoint condition makes the second point a half turn of the first.

(p+x_B)/2=0, (q+y_B)/2=0 ⟹ B=(-p,-q)
2STEP 2

Write down both memberships

Both memberships give two equations in two unknowns.

q=4p²+7p-1, -q=4p²-7p-1
3STEP 3

Add the two equations to find p

Adding cancels the odd part, forcing the first coordinate to plus or minus one half.

0=8p²-2 ⟹ p²=1/4 ⟹ p=±1/2
4STEP 4

Subtract the two equations to find q

Subtracting gives the other coordinate, and the two signs name the same segment.

2q=14p ⟹ q=7p, A=(1/2,7/2), B=(-1/2,-7/2)
5STEP 5

Measure the segment

The legs 1 and 7 give a hypotenuse of 5√2, choice (E).

AB=√((1/2-(-1/2))²+(7/2-(-7/2))²)=√(1+49)=√(50)=5√(2)
Answer
5√2
Because the origin is the midpoint, AB=2 · OA, and OA=√(1/4+49/4)=√(50)/2≈ 3.54, so AB≈ 7.07. Checking that against the choices: 2√(5)≈ 4.47, 5+√(2)/2≈ 5.71 and 5+√(2)≈ 6.41 are all too small. The dangerous one is 7, which is close enough that a rough decimal will not settle it - but 7²=49 while AB²=50, so AB is strictly bigger than 7. Only 5√(2)=√(50) fits, and squaring rather than approximating is what makes the call safe. It is also worth noticing that the answer came out of a very small pair of points: the segment is short and steep, slope 7, which matches the fact that the coefficient 7 never cancelled.
💡Key takeaway

When the origin is the midpoint, the second point is the first one with both signs flipped - so write both conditions and add them, and the terms that survive tell you where the points are.

  • Let the midpoint fix B from A
  • Write down both memberships
  • Add the two equations to find p
  • Subtract the two equations to find q
  • Measure the segment

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