Find the largest value of x for which

x^2 + y^2 = x + y + 4

has a solution, if x and y are real.

Guest Apr 16, 2022

#1**+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.

MaxWong Apr 16, 2022

#1**+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