Coordinates of the centre of a circle, whose radius is 2 unit and which touches the line pair $x^{2}

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Coordinates of the centre of a circle, whose radius is 2 unit and which touches the line pair $x^{2} - y^{2} - 2x + 1 = 0$ are:

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[Key Concept: The pair $x^{2} - y^{2} - 2x + 1 = 0$ is $(x{- 1)}^{2} - y^{2} = 0$ — two lines through $(1,0)$; a radius-2 circle touching both has its centre at distance $2$ from each.]Centre of circles which touches the lines$$\begin{matrix} & x + y = 1	ext{~}	ext{and}	ext{~}x - y = 1 \ & 	ext{~}	ext{are}	ext{~}\ (1 \pm 2\sqrt{2},0)	ext{~}	ext{and}	ext{~} \ & \ (1, \pm 2\sqrt{2}) \end{matrix}$$

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