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

Question

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

Hassan Watkins · Accepted Answer

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        =  kthen       k    +  k  +  2  =  c repesent the level curves f(x,y)=csolving 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 csuch 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.

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

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