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The equation of a circle in the coordinate plane can be written as $Ax^2 + 2y^2 + Bx + Cy = 50.$ The center of the circle is at $(-5,2)$. Let $r$ be the radius of the circle. Find A+B+C+r.

 Feb 15, 2024
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For this to be a circle, A  must =  2

 

So....we have

 

2x^2 + Bx + 2y^2 + Cy = 50     divide through by 2

 

x^2 + (B/2)x + y^2 +(C/2)y  = 25     complete the square on x,y

 

x^2 + (B/2)X + B^2/4 + y^2 + (C/2)y + C^2/4  = 25  + B^2/4 + C^2/4

 

Factor

 

(x + B/2)^2  + ( y + C/2)^2 =  25 + B^2/4 + C^2/4

 

Since the center is  (-5,2)  then

B/2 = 5         C/2 =  -2

B = 10          C = -4

 

The radius^2  =  25 + 10^2/4  + 16/4 =   25 + 25 + 4  = 54

r = sqrt (54)

 

A + B + C + r =   2 + 10 - 4  + sqrt (54)  = 8 + sqrt (54)

 

 

cool cool cool

 Feb 16, 2024

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