Show that x 2 + y 2 = 6is a level curve of f (...

Bruno Thompson

Bruno Thompson

Answered

2022-07-26

Show that x 2 + y 2 = 6
is a level curve of f ( x , y ) = x 2 + y 2 x 2 y 2 + 2

Answer & Explanation

Hassan Watkins

Hassan Watkins

Expert

2022-07-27Added 18 answers

Write f ( x , y ) = x 2 + y 2 x 2 y 2 + 2 as a function of x 2 + y 2
let x 2 + y 2 = k
then k + k + 2 = c repesent the level curves f(x,y)=c
solving the above for k,
k = 1 ± 1 4 ( 2 c ) 2
k = ( 1 ± 1 4 ( 2 c ) 2 ) 2
x 2 + y 2 = ( 1 ± 1 4 ( 2 c ) 2 ) 2
To get the level curve x 2 + y 2 = 6, we simply solve for c
such that ( 1 ± 1 4 ( 2 c ) 2 ) 2 = 6
1 ± 1 4 ( 2 c ) = ± 2 6
1 4 ( 2 c ) = ( 1 ± 2 6 ) 2
c = 1 4 ( 7 + ( 1 ± 2 6 ) 2 ) = c 1 , c 2
f ( x , y ) = c 1 or f ( x , y ) = c 2 repesent the curve x 2 + y 2 = 6.

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