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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

Best Answer 

 #1
avatar+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
avatar+1252 
+1
Best Answer

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

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