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 $AB$ ?

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 $-y_A = 4(-x_A)^2 + 7(-x_A) - 1$ .

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 $\sqrt{ 1^2 + 7^2 } = \sqrt{50} = \boxed{5\sqrt2}$ .