The four distinct points $(0,0),(2,0),(0,-2)$ and $(k,-2)$ are concyclic, if $k$ is equal to

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Aicharya · Accepted Answer

Let the equation of the circle is $ \begin{aligned} & x^2+y^2+2 g x+2 f y+c=0 \ & 	ext { at }(0,0), 0+0+0+0+c=0 \Rightarrow c=0 \ & 	ext { at } \quad(2,0), 4+0+2 g \cdot 2+0+0=0 \ & \Rightarrow \quad g=-1 \ & 	ext { at }(0,-2), 0+4+0-4 f+0=0 \Rightarrow f=1 \ & 	ext { at }(k,-2), k^2+4+2 k g-4 f+0=0 \ & \Rightarrow \quad k^2+4-2 k-4=0 \ & \Rightarrow \quad k(k-2)=0 \ & \Rightarrow \quad k=0, k=2 \ & \end{aligned} $

The four distinct points $(0,0),(2,0),(0,-2)$ and $(k,-2)$ are concyclic, if $k$ is equal to

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