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A circle is centered at (5,15) and has a radius of \(\sqrt{130}\) units. Point Q = (x,y) is on the circle, has integer coordinates, and the value of the x-coordinate is twice the value of the y-coordinate. What is the maximum possible value for x?

 Mar 18, 2020
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 A circle is centered at (5,15) and has a radius of √130 units. Point Q = (x,y) is on the circle, has integer coordinates, and the value of the x-coordinate is twice the value of the y-coordinate. What is the maximum possible value for x? 

 

We  have  this equation

 

(x - 5)^2 + (y - 15)^2  =  130

 

Let the point we seek  be  ( 2y, y)     

 

So we have

 

 

(2y -5)^2   + ( y - 15)^2  =  130        simplify

 

4y^2  - 20y + 25  + y^2  - 30y + 225   =  130

 

5y^2   - 50y + 250  = 130

 

5y^2  - 50y +  120  =  0        divide through  by  5

 

y^2  - 10y  + 24  = 0            factor

 

(y - 6) ( y - 4)  =  0

 

Setting each factor to 0   and solving for   we  get that

 

y  = 6                  or              y = 4

 

So...it's obvious that x is maximized when  y =  6....and x  = 12

 

See the graph here : https://www.desmos.com/calculator/bkoknegsvd

 

 

cool cool cool

 Mar 18, 2020

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