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# Helps!

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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.

Jun 8, 2024

#1
+759
+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! :)

Jun 8, 2024

#1
+759
+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