Circle through the intersection of two circles — A/L Combined Mathematics

The equation of any circle passing through the points of intersection of a given circle S ≡ x² + y² + 2gx + 2fy + c = 0 and a straight line U ≡ ax + by + k = 0 is represented by the family S + λU = 0, where λ is a real parameter. Similarly, for two intersecting circles S₁ = 0 and S₂ = 0, the equation of the common chord (radical axis) is S₁ - S₂ = 0 (a linear equation since x² and y² cancel), and the family of circles passing through their intersection points is given by S₁ + λ(S₁ - S₂) = 0 or S₁ + λS₂ = 0 (where λ ≠ -1).

Check before you answer (A/L Pure Maths Paper 1)

1. Parameter Determination: Setting up S + λU = 0 and applying a given third condition (e.g., passing through a point, or having its centre on a specified line) to solve for λ.
2. Cancellation of Second-Degree Terms: Ensuring the radical axis S₁ - S₂ = 0 eliminates both x² and y² terms yielding a linear equation 2(g₁ - g₂)x + 2(f₁ - f₂)y + (c₁ - c₂) = 0.
3. Orthogonal Intersection: Applying the orthogonality condition to the family equation when finding circles in coaxial systems.