Question

Find the equations of the circles passing through the points of intersection of the circles x² + y² –2x – 4y – 4 = 0 and x² + y² – 10x – 12y + 40 = 0 and whose radius is 4.

Answer 1

Any circle through the intersection of given circles is S₁ + λS₂ = 0 

or (x² + y² – 2x – 4y – 4) + l(x² + y² – 10x – 12y + 40 ) = 0

16(1 + 2λ + λ²) = 1 + 10λ + 25λ² + 4 + 24λ + 36λ² – 40λ 2 – 40λ + 4 + 4λ 

or 16 + 32λ + 16λ² = 21λ² – 2λ + 9 

or 5λ² – 34λ – 7 = 0 

∴ (λ – 7) (5λ + 1) = 0 ∴ λ = 7, – 1/5 

Putting the values of λ in (i) the required circles are 

2x² + 2y² – 18x – 22y + 69 = 0 and x² + y² – 2y – 15 = 0