+0

# Algebra

0
2
1
+174

Find the largest value of x for which
x^2 + y^2 = 4x + 5y
has a solution, if x and y are real.

Jun 12, 2024

#1
+1252
+1

Notice that $$x^2 + y^2 = 4x + 5y$$ is very similar to the equation of a circle.

We have $$x^2-4x+y^2-5y=0$$

Now, completing the square for both x and y, we get

$$(x-2)^2+(y-5/2)^2=25/4+16$$

This is the official equation of a circle.

It has the center at $$(2, 5/2)$$ and a radius of $$\sqrt{89/4}$$

The largest value of x is basically the x value of the center added to the radius.

Thus, we have $$2+\frac{\sqrt{89}}{2} = \frac{4+\sqrt{89}}{2}$$

So our final answer is $$\frac{4+\sqrt{89}}{2}$$

Thanks! :)

Jun 12, 2024

#1
+1252
+1

Notice that $$x^2 + y^2 = 4x + 5y$$ is very similar to the equation of a circle.

We have $$x^2-4x+y^2-5y=0$$

Now, completing the square for both x and y, we get

$$(x-2)^2+(y-5/2)^2=25/4+16$$

This is the official equation of a circle.

It has the center at $$(2, 5/2)$$ and a radius of $$\sqrt{89/4}$$

The largest value of x is basically the x value of the center added to the radius.

Thus, we have $$2+\frac{\sqrt{89}}{2} = \frac{4+\sqrt{89}}{2}$$

So our final answer is $$\frac{4+\sqrt{89}}{2}$$

Thanks! :)

NotThatSmart Jun 12, 2024