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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
avatar+120966 
+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

 

 

cool cool cool

 Jun 27, 2021
 #2
avatar+708 
+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

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