+0  
 
0
297
2
avatar

Let x and y be real numbers such that x^2 + y^2 = 4(x + y). Find the largest possible value of x.

 Jun 2, 2020
 #1
avatar+26735 
+1

x^2 + y^2 = 4(x + y).=

x^2 -4x       + y^2-4y      = 0

(x-2)^2        + ( y-2)^2    =  8           this is a circle with center at  2,2    and   r = sqrt 8       max will be    2 + sqrt 8

 Jun 2, 2020
 #2
avatar+21953 
0

x2 + y2  =  4(x + y)     is an equation of a circle.

 

x2 + y2  =  4(x + y)      --->                                   x2 + y2  =  4x + 4y

                                             (x2 - 4x      ) + (y2 - 4y      )  =  0

Complete the squares:        (x2 - 4x + 4) + (y2 - 4y + 4)  =  8

Factor:                                                 (x - 2)2 + (y - 2)2  =  8

 

This circle has its center at  (2, 2)  and has a radius of  sqrt(8).

 

The largest x-value occurs at the right end of the circle  --  the x-value of the center plus the radius  -- 2 + sqrt(8)

 Jun 2, 2020

13 Online Users