If P and Q are the points of intersection of the circles x² + y² + 3x + 7y + 2p = 0 and x² + y² + 2x + 2y - p² = 0, then there is a circle passing through P, Q and (1,1) for all values of p all except two values of p all except one value of p exactly one value of p

S₁ = x² + y² + 3x + 7y + 2p = 0
S₂ = x² + y² + 2x + 2y - p² = 0
S₁ - S₂ will give equation of radical axis
Radical axis is x + 5y + p² + 2p = 0
Required Equation of circle will be S₁ + λ[S₁ - S₂]
x² + y² + 3x + 7y + 2p + λ[x + 5y + p² + 2p] = 0 (i)
(i) passes through (1,1)
λ = -(2p + 7)/(p + 1)² (p ≠ -1)
Hence, option 'B' is correct.