+981 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?
+128408 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
