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(A) The equation of the common chord (radical axis) of the two circles $S_1: x^2 + y^2 + 3x + 7y + 2p - 5 = 0$ and $S_2: x^2 + y^2 + 2x + 2y - p^2 = 0

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all except one value of $p$

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(A) The equation of the common chord (radical axis) of the two circles $S_1: x^2 + y^2 + 3x + 7y + 2p - 5 = 0$ and $S_2: x^2 + y^2 + 2x + 2y - p^2 = 0$ is given by $S_1 - S_2 = 0$.$(x^2 + y^2 + 3x + 7y + 2p - 5) - (x^2 + y^2 + 2x + 2y - p^2) = 0$$x + 5y + 2p - 5 + p^2 = 0 \quad \dots(i)$For a circle to pass through the intersection points $P$ and $Q$ and the point $(1, 1)$,the point $(1, 1)$ must not lie on the common chord $PQ$.Substituting $(1, 1)$ into equation $(i)$:$1 + 5(1) + 2p - 5 + p^2 
eq 0$$p^2 + 2p + 1 
eq 0$$(p + 1)^2 
eq 0$$p 
eq -1$Thus,there exists a circle passing through $P, Q$ and $(1, 1)$ for all values of $p$ except $p = -1$.

If $P$ and $Q$ are the points of intersection of the circles $x^2 + y^2 + 3x + 7y + 2p - 5 = 0$ and $x^2 + y^2 + 2x + 2y - p^2 = 0$,then there is a circle passing through $P, Q$ and $(1, 1)$ for:

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