What is the equation of the circle that passes through the origin, has its center on the line \(x+y=4,\) and cuts the circle \(x^2+y^2-6x+16y+24=0\) orthogonally?

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Find the sum of all possible values of \( b \) such that the two points \( (4, 0 ) \) and \( (- 4, 0) \) lie on the circle \( x^2 + y^2 + ax + by +c =0 \) with radius \( 6 \) .

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If the circle \(x^2+y^2+Ax+By+C=0\) passes through the origin and has the same center as the circle \(x^2+y^2-9x+8y-3=0,\) what is the value of \(A+B+C?\)

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