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.

BRAINBOLT Jun 8, 2024

#1**+1 **

Let's first move all terms to one side of the equation and combine all like terms.

We get

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

Now, let's complete the square for x and y on the left side the equation.

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

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

This is the equation for a circle.

According to the circle rules, this circle has a center at \((21/2 , -13/2)\)and radius \(\sqrt{355 / 2}\)

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

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

Thanks! :)

NotThatSmart Jun 8, 2024

#1**+1 **

Best Answer

Let's first move all terms to one side of the equation and combine all like terms.

We get

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

Now, let's complete the square for x and y on the left side the equation.

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

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

This is the equation for a circle.

According to the circle rules, this circle has a center at \((21/2 , -13/2)\)and radius \(\sqrt{355 / 2}\)

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

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

Thanks! :)

NotThatSmart Jun 8, 2024