+0  
 
0
56
1
avatar

Find the largest value of $x$ for which \[x^2 + y^2 = x + y\]has a solution, if $x$ and $y$ are real.

 Dec 2, 2020
 #1
avatar+114592 
+1

Rearange  as

 

x^2   - x  +  y^2   - y    =   0      complete the square on x and  y

 

x^2  - x  +1/4    +  y^2  - y + 1/4 =   1/4  +  1/4       factor

 

(x - 1/2)^2  +  ( y -1/2)^2   =   1/2

 

x will be maximized  when  y = 1/2

 

So

( x - 1/2)^2   +  (1/2 - 1/2)^2  =  1/2

 

(x  -1/2)^2    +   0     =   1/2

 

(x - 1/2)^2  = 1/2           take the positive root

 

x - 1/2 =  sqrt (1/2)

 

x - 1/2  =  sqrt (2) / 2

 

x =  [ 1 + sqrt (2)  ]  / 2

 

 

cool cool cool

 Dec 2, 2020

46 Online Users