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Find the largest value of x for which
x^2 + y^2 = x + y + 4
has a solution, if x and y are real.

 Apr 16, 2022

Best Answer 

 #1
avatar+9459 
+1

We visualize the equation as a circle on the coordinate plane.

Graph: https://www.desmos.com/calculator/swjzoksjmi

 

The circle is centered at \(\left(\dfrac12, \dfrac12\right)\) and has a radius \(\dfrac{3\sqrt 2}2\). Therefore, for the largest value of x, we find the maximum possible x-coordinate on the circle, i.e., the x-coordinate of the right-most point.

 

\(\max x = \dfrac12 + \text{radius of circle} = \dfrac{1 + 3\sqrt 2}2\)

 

Further explanations are typed in the graph.

 Apr 16, 2022
 #1
avatar+9459 
+1
Best Answer

We visualize the equation as a circle on the coordinate plane.

Graph: https://www.desmos.com/calculator/swjzoksjmi

 

The circle is centered at \(\left(\dfrac12, \dfrac12\right)\) and has a radius \(\dfrac{3\sqrt 2}2\). Therefore, for the largest value of x, we find the maximum possible x-coordinate on the circle, i.e., the x-coordinate of the right-most point.

 

\(\max x = \dfrac12 + \text{radius of circle} = \dfrac{1 + 3\sqrt 2}2\)

 

Further explanations are typed in the graph.

MaxWong Apr 16, 2022

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