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If $P$ and $Q$ are the points of intersection of the circles $x^2+y^2+3 x+7 y+2 p-5=0$ and $x^2+y^2+2 x+2 y-p^2=0$, then there is a circle passing through $P, Q$ and $(1,1)$ for
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The correct answer is:
all values of $p$
all values of $p$
Given circles $S=x^2+y^2+3 x+7 y+2 p-5=0$
$$
S^{\prime}=x^2+y^2+2 x+2 y-p^2=0
$$
Equation of required circle is $S+\lambda S^{\prime}=0$
As it passes through $(1,1)$ the value of $\lambda=\frac{-(7+2 p)}{\left(6-p^2\right)}$
If $7+2 p=0$, it becomes the second circle
$\therefore$ it is true for all values of $p$
$$
S^{\prime}=x^2+y^2+2 x+2 y-p^2=0
$$
Equation of required circle is $S+\lambda S^{\prime}=0$
As it passes through $(1,1)$ the value of $\lambda=\frac{-(7+2 p)}{\left(6-p^2\right)}$
If $7+2 p=0$, it becomes the second circle
$\therefore$ it is true for all values of $p$
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