Processing math: 100%
 
+0  
 
+2
17
1
avatar+280 

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

Best Answer 

 #1
avatar+1944 
+1

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

We get

x221x+y2+13y=25

 

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

x221x+441/4+y2+13y+169/4=25+441/4+169/4

(x21/2)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 355/2

 

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

 

We have x=(21/2)+355/2

 

Thanks! :)

 Jun 8, 2024
 #1
avatar+1944 
+1
Best Answer

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

We get

x221x+y2+13y=25

 

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

x221x+441/4+y2+13y+169/4=25+441/4+169/4

(x21/2)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 355/2

 

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

 

We have x=(21/2)+355/2

 

Thanks! :)

NotThatSmart Jun 8, 2024

1 Online Users

avatar