Let \((x,y)\) be an ordered pair of real numbers that satisfies the equation \(x^2+y^2=14x+48y\). What is the minimum value of x?
x^2 + y^2 = 14x + 48y rearrange as
x^2 - 14x + y^2 - 48y = 0 complete the square on x and y
x^2 -14x + 49 + y^2 - 48y + 576 = 49 + 576 factor and simplify
(x -7)^2 + ( y - 24)^2 = 625
This is a circle centered at ( 7 , 24) with a radius of √625 = 25
The minimum value for x will be : x coordinate of the center minus the radius =
7 - 25 = -18