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Points Yielding Minimum Perimeter of 25

Source:

August 9, 2024

perimeteroptimizationcalculus2022

Problem Statement

Let

P = (-4, 0)

and

Q = (4, 0)

be two points on the

x

-axis of the Cartesian coordinate plane, and let

X

and

Y

be points on the

x

-axis and

y

-axis, respectively, such that over all

Z

on line

\overleftrightarrow{XY}

, the perimeter of

\triangle ZPQ

has a minimum value of

25

. What is the smallest possible value of

XY^2

?

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