Determining all ( a , b ) on the unit circle such that 2 x + 3 y + 1 &#

Jameson Lucero 2022-06-29 Answered
Determining all ( a , b ) on the unit circle such that 2 x + 3 y + 1 a ( x + 2 ) + b ( y + 3 ) for all ( x , y ) in the unit disk
In the middle of another problem, I came up with the following inequality which needed to be solve for ( a , b ) :
2 x + 3 y + 1 a ( x + 2 ) + b ( y + 3 )
for all ( x , y ) R 2 with x 2 + y 2 1..
Here the solution for ( a , b ) must be a subset of the unit circle, and I believe that it is a singleton.
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Answers (1)

Yair Boyle
Answered 2022-06-30 Author has 10 answers
Rewrite 2 x + 3 y + 1 a ( x + 2 ) + b ( y + 3 ) as ( 2 a ) x + ( 3 b ) y 2 a + 3 b 1.
Solve for the intersection of the line ( 2 a ) x + ( 3 b ) y = 2 a + 3 b 1 with x 2 + y 2 = 1 to get that the x-coordinates of the intersection are
2 a 2 3 a b + 5 a + 6 b 2 ± ( b 3 ) 2 ( 3 a 2 + 12 a b + 8 b 2 12 ) a 2 4 a + b 2 6 b + 13 .
If this has two real solutions, then the inequality cannot be satisfied for all ( x , y ) in the unit circle, so we need to find the intersection of ( b 3 ) 2 ( 3 a 2 + 12 a b + 8 b 2 12 ) 0 and a 2 + b 2 1. Simultaneously solving these equations, we find that the real solutions for ( a , b ) are ( 2 13 , 3 13 ) and ( 2 13 , 3 13 ). It's easy to see that the first solution is valid and the second is invalid - plug in ( x , y ) = ( 0 , 0 ), for instance.
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