If the equation of a circle is x2 + y2 + Ax + By + C = 0 and the equation of a line is Y = mx + n, substituting the line's equation into the circle's equation produces a quadratic equation of the form (1 + m2)x2 + (A + 2mn + Bm)x + n2 + Bn + C = 0. The discriminant of this quadratic equation, D = b2 - 4ac, determines whether the line and circle have: two intersection points if D > 0, one intersection point if D = 0, or no intersection points if D < 0.
1. If a circle have an equation of circle : And a line have an equation of line : x 2 + y 2 + Ax + By + C = 0 Y = mx + n
2. Suppose that a line with equation of line y = mx + n and a circle with equation of x 2 + y 2 + Ax + By + C = 0 , then substitute the line’s equation into the circle’s equation to produce a new equation in the form of quadratic equation like above : x 2 + (mx + n) 2 + Ax + B (mx + n) + C = 0 x 2 + (m 2 x 2 + 2mnx + n 2 ) + Ax + B mx +Bn + C = 0 x 2 + m 2 x 2 + 2mnx + Ax + Bmx + n 2 + Bn + C = 0 (1 + m 2 ) x 2 + (2mn + A + Bm)x + n 2 + Bn + C = 0 Atau (1 + m 2 ) x 2 + (A + 2mn + B m )x + n 2 + Bn + C = 0
3.
4. The line and circle have two intersection points The line and circle have an exactly one intersection point The line and circle have no intersection point
