Family of circles:

(i) If S ≡ x² + y² + 2gx + 2fy + c = 0 and S' ≡ x² + y² + 2g'x + 2f'y + c' = 0 are 2 intersecting circles, then the family of circles passing through point of intersection of S and S' can be given by S + λS' = 0, (here λ is a parameter λ ≠ -1).

(ii) If S ≡ x² + y² + 2gx + 2fy + c = 0 is a circle which is intersected by straight line L=λx+my+n=0 at 2 real and distinct points, then the equation of the family of circles passing through the point of intersection of the circle and the given line can be given by S + λL = 0.(herel is a parameter).

(iii) The equation of family of circles passing through 2 given points (x₁, y₁) and (x₂, y₂) is written in the form.

                        (x - x₁)(x - x₂) + (y - y₁)(y - y₂) +  =0  (where λ is a parameter).

(iv) The equation of family of circles which touch the line y - y₁ = m(x - x₁)  at (x₁, y₁) for any value of m is (x - x₁)² + (y - y₁)² + λ[(y - y₁) -m(x - x₁)] = 0. If the value of m is infinite, the equation is (x - x₁)² + (y - y₁)² + l(x - x₁) = 0.

Illustration: The equation of the common chord of 2 circles is x+ y = 1. One of the circles has the ends of a diameter at points (1 , -3) and (4, 1) and the other passes through point (1 ,2). Find the equation of the 2 circles

 Solution:  Let c₁ is the circle whose end point of the diameter are given by

                ∴  c₁  ≡ (x - 1) (x - 4) + (y +3) ( y - 1)  = 0

                 ≡ x² + y² - 5x + 2y + 1 = 0  

                 Let c₂ is another circles

                 ∴ The equation of family of circles passing through the points of intersection of the circle c₁ and the given line

                  x² + y² - 5x + 2y +1 + l(x + y - 1) = 0 ................(1)

                   for c₂ , 1^st equation satisfy point (1 ,2 )

                  ∴  λ = -5/2 

                  ∴  c₂ = 2x² + 2y² - 15x - y + 7 = 0 

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