Family of Circles

Family of Circles

Let S = x² + y² + 2gx + 2fy + c= 0

S′ = x² + y² + 2g′x + 2f′y + c′ = 0 and L = px + qy + r = 0 then

1. If S = 0 and S′ = 0 intersects in real and distinct points S + λS = 0 (λ≠ -1) represents a (Family of Circles) passing through these points. S - S′ = 0 for λ = -1 represents a common chord of the circles S = 0 and S′ = 0.
2. S = 0 and S′ = 0 touch each other then S - S′ = 0 is the equation of the common tangent to the two circles at the point of contact.
3. If S = 0 and L = 0 intersect in two real distinct points, then S + λL = 0 represents a family of circles passing through these points.
4. If L = 0 is a tangent to the circle S = 0 at P then S + λL = 0 represents a family of circles touching S = 0 at P having L = 0 as the common tangent at P.