Geometry
Conic Sections
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?
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 \( 6 \) .
If the line \(y=ax+b\) cuts the circle \(x^2+y^2-4x+8y-29=0\) into two equal parts, then what is the value of \(2a+b ?\)
If the circle \[x^2+y^2+ax+by+c=0\] passes through three points \[(-1,-8), (2,1), (6,-1),\] what is the value of \(a+b+c ?\)
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 \(A+B+C?\)