Problem:
Points A and B are on the parabola y=4x2+7x−1, and the origin is the midpoint of AB. What is the length of AB ?
Answer Choices:
A. 25
B. 5+22
C. 5+2
D. 7
E. 52
Solution:
Let B=(a,b) and A=(−a,−b). Then
4a2+7a−1=b and 4a2−7a−1=−b
Subtracting gives b=7a, so 4a2+7a−1=7a. Thus
a2=41 and b2=(7a)2=449
SO
AB=2a2+b2=2450=52
The problems on this page are the property of the MAA's American Mathematics Competitions