Conic Sections
Circle General equation of a circle with the center S(p, q) - translated circle
Equation of the circle with the center at the origin O(0, 0) Circle
General equation of a circle with the center S(p, q) - translated circle
A circle with the center at the point S(p, q) and radius r is a set of all points P(x, y) of a plane to whom the
 distance from the center,   SP = r  or     is the general equation of a circle with the center S(p, q).
 Equation of the circle with the center at the origin O(0, 0),  Example:  A circle passes through points A(2, 4) and B(-2, 6) and its center lies on a line x + 3y - 8 = 0.
Find equation of the circle.
Solution:   The intersection of the chord AB  bisector and the given line is the center S of the circle, since the bisector is normal through the midpoint M, then As the bisector is perpendicular to the line AB Equation of the bisector M(0, 5) and mn = 2  =>   y - y1 = m( x - x1), gives       y - 5 = 2(x - 0)   or   -2x + y - 5 = 0.   So, the equation of the circle,     (x - p)2 + (y - q)2 = r2   =>    (x + 1)2 + (y - 3)2 = 10.
Circle through three points
A circle is uniquely determined by three points not lying on the same line. If given are points A, B and C then the intersection of any pair of the perpendicular bisectors of the sides of the triangle ABC is the center of the circle.
 Since all three points lie on the circle, their coordinates must satisfy equation of the circle (x - p)2 + (y - q)2 = r2. Thus, we obtain the system of three equations in three unknowns p, q and r. Subtracting second equation from first and then third from first we obtain two equations in two unknowns p and q. Solutions of that system plug into any of three equations to get r. Example:  Find equation of a circle passing through three points, A(-2, -6), B(5, -7) and C(6, 0).
Solution:   The coordinates of the points, A, B and C plug into equation  (x - p)2 + (y - q)2 = r2,  Thus, the equation of the circle through points A, B and C,    (x - 2)2 + (y + 3)2 = 25.   Intermediate algebra contents 