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

x^2 + y^2 = 4x - 8y + 17x - 5y + 25,

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

BRAlNBOLT May 29, 2024

#1**+1 **

First of all, let's move almost all the variables to one side of the equation and combine all the like terms.

We get \(x^2 - 21x + y^2 + 13y = 25 \).

Now, we complete the square for both x and y on the right side of the equation.

\(x^2 - 21x + 441/4 + y^2 + 13y + 169/4 = 25 + 441/4 + 169/4 \)

\((x - 21/2)^2 + (y + 13/2)^2 = 355 / 2 \)

Wait! This looks like the equation for a circle!

Indeed, we have a circle at center \((21/2 , -13/2) \) with radius \(\sqrt{355 / 2}\).

The value of the largest possible x is just the radius added onto the x value of the center.

\( x = (21/2) + \sqrt{355 / 2}\)

Thanks! :)

NotThatSmart May 29, 2024

#1**+1 **

Best Answer

First of all, let's move almost all the variables to one side of the equation and combine all the like terms.

We get \(x^2 - 21x + y^2 + 13y = 25 \).

Now, we complete the square for both x and y on the right side of the equation.

\(x^2 - 21x + 441/4 + y^2 + 13y + 169/4 = 25 + 441/4 + 169/4 \)

\((x - 21/2)^2 + (y + 13/2)^2 = 355 / 2 \)

Wait! This looks like the equation for a circle!

Indeed, we have a circle at center \((21/2 , -13/2) \) with radius \(\sqrt{355 / 2}\).

The value of the largest possible x is just the radius added onto the x value of the center.

\( x = (21/2) + \sqrt{355 / 2}\)

Thanks! :)

NotThatSmart May 29, 2024