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# Algebra

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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.

May 29, 2024

#1
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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! :)

May 29, 2024

#1
+806
+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