Let $C$ be the circle with equation $x^2+2y-9=-y^2+18x+9$. If $(a,b)$ is the center of $C$ and $r$ is its radius, what is the value of $a+b+r$?
+128475 x^2 + 2y - 9 = -y^2 + 18x + 9 rearrange as
x^2 - 18x + y^2 + 2y = 18 complete the square on x and y
Take (1/2) 18 = 9 ....square it =81 and add to both sides
Take (1/2)2 = 1 square it = 1 and add to both sides
x^2 - 18x + 81 + y^2 + 2y + 1 = 18 + 81 + 1 factor the left and simplify the right
(x - 9)^2 + ( y + 1)^2 = 100
a = 9 b = -1 sqrt (100) = r =10
a + b + r =
9 - 1 + 10 =
18
+874 Image: https://ibb.co/2Z1yrj5
$x^2+2y-9=-y^2+18x+9$
$x^2 -18x + y^2 +2y - 18 = 0$
$(x-9)^2 - 81 + (y+1)^2 - 1 - 18 = 0$
$(x-9)^2 + (y+1)^2 - 100 = 0$
$(x-9)^2 + (y+1)^2 = 100$
$(x-9)^2 + (y+1)^2 = 10^2$
$(x, y, r) = (9, -1, 10)$
$9 + (-1) + 10 = \boxed{18}$
