The equation of a family of circles passing through the intersection of circles x² + y² + 2gx + 2fy + c = 0 and line L = lx + my + n = 0 is x² + y² + 2gx + 2fy + c + λ (lx + my + n) = 0 i.e., S + λ L = 0 where λ is any real.

The equation of family of circles passing through the points A (x₁, y₁) and B (x₂, y₂) is (x – x₁) (x – x₂) + (y – y₁) (y – y₂).

(Or)

(x – x₁) (x – x₂) + (y – y₁) (y – y₂) + λL = 0.

Where L = 0 represents a line passing through A (x₁, y₁), B (x₂, y₂).

In the above equation if x₂ → x₁ and y₂ → y₁, then it reduces to (x – x₁)² + (y – y₁)² + λL = 0 also when x₂ → x₁ and y₂ → y₁, the equation of line AB becomes tangent to circle S = 0.

It follows from that equation (x – x₁)² + (y – y₁)² + λL = 0, λ ϵ R represents a family of circles touching L = 0 at (x₁, y₁).

The equation of family of circles touching the circle S = x² + y² + 2gx + 2fy + c = 0 at point P (x₁, y₁) is S + λL = 0.

L = xx₁ + yy₁ + g (x + x₁) + f (y + y₁) + c = 0.

The equation of family of circles passing through the intersection of circles is S₁ + λS₂ = 0 Where (λ ≠ – 1) is any real.