¶ Spring 2021 AMC 12A Problem 20
Problem:
Suppose that on a parabola with vertex and a focus there exists a point such that and . What is the sum of all possible values of the length ?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Let be the directrix of such a parabola. By definition, the parabola is the set of points such that the distance from to is equal to . Let and be the orthogonal projections of and , respectively, onto , and let and be the orthogonal projections of and , respectively, onto line . Because , there are two possible configurations that may arise, and they are shown below.
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Let . Because , it follows that and . Because is a rectangle, , so applying the Pythagorean Theorem to and gives
This equation simplifies to , which has solutions
Specifically, the lesser solution is the value of with the right configuration and the greater solution is the value of with the left configuration. The requested answer is .
OR
Let be the origin, let lie on the positive -axis, and let . The equation of the parabola is then . If are the coordinates of , then and . Substituting for and gives , which simplifies to , and the solution proceeds as above.
The problems on this page are the property of the MAA's American Mathematics Competitions