Let $x$ and $y$ be real numbers. If $x$ and $y$ satisfy

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

then find the largest possible value of $x.$ Give your answer in exact form using radicals, simplified as far as possible.

RedDragonl May 28, 2024

#1**+1 **

First off, let's move all the terms to one side of the equation before combining like terms!

\(x^2 - 4x + y^2 - 2y = 0 \)

Completing the square for x and y, we get

\(x^2 - 4x + 4 + y^2 - 2y + 1 = 4 +1 \\ (x- 2)^2 + ( y - 1)^2 = 5\)

Wait! This is the equation for a circle.

Using the equations for circles, this circle has a radius of \(\sqrt{5}\) and center (2, 1).

The largest value of x would be \(x = 2 +\sqrt 5\).

The reason for this is because the largest value of x would just be the radius plus the x value of the center!

Thanks! :)

NotThatSmart May 28, 2024

#1**+1 **

Best Answer

First off, let's move all the terms to one side of the equation before combining like terms!

\(x^2 - 4x + y^2 - 2y = 0 \)

Completing the square for x and y, we get

\(x^2 - 4x + 4 + y^2 - 2y + 1 = 4 +1 \\ (x- 2)^2 + ( y - 1)^2 = 5\)

Wait! This is the equation for a circle.

Using the equations for circles, this circle has a radius of \(\sqrt{5}\) and center (2, 1).

The largest value of x would be \(x = 2 +\sqrt 5\).

The reason for this is because the largest value of x would just be the radius plus the x value of the center!

Thanks! :)

NotThatSmart May 28, 2024