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There exists a circle C with radius 5, passing through the point (3,2), such that the center of C lies on the line x=7 and has positive y-coordinate. C is the graph of the equation x^2 + ax + y^2 + by + c = 0, where a,b,c, are integers (not necessarily positive). Find a+b+c.

 

i think the answer might be 25, not sure tho

 Aug 12, 2023
 #5
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+1

Since it passes through   (3,2)  with the center at  (7,y) we have

 

(3 - 7)^2 + ( 2 - y)^2 =  5^2

 

16 +  (2 - m)^2  = 25

 

(2 - y)^2  =   25 -16

 

(2 - y)^2 =  9       take the NEGATIVE square root

 

2 - y  =  -3          rearrange  as

 

5  =  y

 

The center is  ( 7, 5)    and  we  have the equation of a  circle

 

(x - 7)^2 + ( y - 5)^2 =  25

 

x^2 -14y + 49 + y^2 - 10y + 25 =  25

 

x^2 - 14y + y^2 -10y + 49 =  0

 

a = -14    b = -10     c = 49      and their  sum  =  25

 

And you are correct !!!!

 

cool cool cool

 Aug 13, 2023

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