# 2004 AMC 12B Problems/Problem 18

## Problem

Points $A$ and $B$ are on the parabola $y=4x^2+7x-1$, and the origin is the midpoint of $AB$. What is the length of $AB$? $\mathrm{(A)}\ 2\sqrt5 \qquad \mathrm{(B)}\ 5+\frac{\sqrt2}{2} \qquad \mathrm{(C)}\ 5+\sqrt2 \qquad \mathrm{(D)}\ 7 \qquad \mathrm{(E)}\ 5\sqrt2$

## Solution

Let the coordinates of $A$ be $(x_A,y_A)$. As $A$ lies on the parabola, we have $y_A=4x_A^2+7x_A-1$. As the origin is the midpoint of $AB$, the coordinates of $B$ are $(-x_A,-y_A)$. We need to choose $x_A$ so that $B$ will lie on the parabola as well. In other words, we need $-y_A = 4(-x_A)^2 + 7(-x_A) - 1$.

Substituting for $y_A$, we get: $-4x_A^2 - 7x_A + 1 = 4(-x_A)^2 + 7(-x_A) - 1$.

This simplifies to $8x_A^2 - 2 = 0$, which solves to $x_A = \pm 1/2$. Both roots lead to the same pair of points: $(1/2,7/2)$ and $(-1/2,-7/2)$. Their distance is $\sqrt{ 1^2 + 7^2 } = \sqrt{50} = \boxed{5\sqrt2}$.

## Alternate Solution

Let the coordinates of $A$ and $B$ be $(x_A, y_A)$ and $(x_B, y_B)$, respectively. Since the median of the points lies on the origin, $x_A + x_B = y_A + y_B = 0$ and expanding $y_A + y_B$, we find: $$4x_A^2 + 7x_A - 1 + 4x_B^2 + 7x_B - 1 = 0$$ $$4(x_A^2 + x_B^2) + 7(x_A + x_B) = 2$$ $$x_A^2 + x_B^2 = \frac{1}{2}.$$

It also follows that $(x_A + x_B)^2 = 0$. Expanding this, we find: $$x_A^2 + 2x_A x_B + x_B^2 = 0$$ $$\frac{1}{2} + 2x_A x_B = 0$$ $$x_A x_B = -\frac{1}{4}.$$

To find the distance between the points, $\sqrt{(x_A - x_B)^2 + (y_A - y_B)^2}$ must be found. Expanding $y_A - y_B$: $$y_A - y_B = 4x_A^2 + 7x_A - 1 - 4x_B^2 - 7x_B + 1$$ $$= 4(x_A^2 - x_B^2) + 7(x_A - x_B)$$ $$= 4(x_A + x_B)(x_A - x_B) + 7(x_A - x_B)$$ $$= 7(x_A - x_B)$$ we find the distance to be $\sqrt{50(x_A - x_B)^2}$. Expanding this yields $5\sqrt{2(x_A^2 + x_B^2 - 2x_A x_B)} = \boxed{5\sqrt{2}}$.

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