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The locus of the centers of the circles that are passing through the intersection of the circles $x^2+y^2=1$ and $x^2+y^2-2 x+y=0$ is
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a line whose equation is x + 2 y = 0.
Equation of the circles passing through the intersection of the circles $x^2+y^2=1$ and
$\begin{aligned} & x^2+y^2-2 x+y=0 \text { is } \\ & \quad\left(x^2+y^2-1\right)+k\left(x^2+y^2-2 x+y\right)=0 \\ & \Rightarrow x^2+y^2-\left(\frac{2 k}{k+1}\right) x+\left(\frac{k}{k+1}\right) y-1=0\end{aligned}$
On comparing with general form
$\begin{gathered}x^2+y^2+2 g x+2 f y+c=0 \\ g=\frac{-k}{k+1}, f=\frac{k}{2(k+1)} \text { and } c=-1 \\ \therefore \text { Centre }=(-g,-f)=\left(\frac{k}{k+1}, \frac{-k}{2(k+1)}\right) \\ \text { If } x=\frac{k}{k+1} \text { and } y=\frac{-k}{2(k+1)} \Rightarrow x+2 y=0\end{gathered}$
$\begin{aligned} & x^2+y^2-2 x+y=0 \text { is } \\ & \quad\left(x^2+y^2-1\right)+k\left(x^2+y^2-2 x+y\right)=0 \\ & \Rightarrow x^2+y^2-\left(\frac{2 k}{k+1}\right) x+\left(\frac{k}{k+1}\right) y-1=0\end{aligned}$
On comparing with general form
$\begin{gathered}x^2+y^2+2 g x+2 f y+c=0 \\ g=\frac{-k}{k+1}, f=\frac{k}{2(k+1)} \text { and } c=-1 \\ \therefore \text { Centre }=(-g,-f)=\left(\frac{k}{k+1}, \frac{-k}{2(k+1)}\right) \\ \text { If } x=\frac{k}{k+1} \text { and } y=\frac{-k}{2(k+1)} \Rightarrow x+2 y=0\end{gathered}$
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