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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\$?

Jun 27, 2021

#1
+1

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   Jun 27, 2021
#2
+1

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}\$

Jun 27, 2021