Let ( c 1 </msub> , c 2 </msub> ) be a fixed point in

Janet Forbes

Janet Forbes

Answered question

2022-07-08

Let ( c 1 , c 2 ) be a fixed point in R 2 . How to maximize | x 1 x 2 c 1 c 2 | subject to the condition that ( x 1 c 1 ) 2 + ( x 2 c 2 ) 2 < 1. i.e.
sup ( x 1 , x 2 ) R 2 : ( x 1 c 1 ) 2 + ( x 2 c 2 ) 2 < 1 | x 1 x 2 c 1 c 2 | = ?
Note : This is a generalization of the problem of maximizing | x 1 x 2 | subject to the condition x 1 2 + x 2 2 < 1. I'm stuck with it. Any help would be much appreciated.

Answer & Explanation

Elias Flores

Elias Flores

Beginner2022-07-09Added 24 answers

The allowable region for ( x 1 , x 2 ) is a unit circle centered on ( c 1 , c 2 ). The optimal point will be one where the hyperbola x 1 x 2 = k is tangent to the circle and k will be the maximum. It will be on the circle, so we can write x 1 = c 1 + cos θ , x 2 = c 2 + sin θ. Then x 1 x 2 c 1 c 2 = c 1 sin θ + c 2 cos θ + sin θ cos θ. If ( c 1 , c 2 ) is in the first quadrant we want θ in the first quadrant as well. We can now take a derivative with respect to theta and set it to zero, getting
c 1 cos θ c 2 sin θ + cos 2 θ sin 2 θ = 0
Now solve for θ

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