+0 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.
+129771 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)
