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(B) Let the centre of the circle be $(h, k)$ and its radius be $r$. Since the circle touches the $x$-axis,the radius $r = |k|$.Since the circle touche

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$\{ (x, y) : {x^2} = 4y\} \cup \{ (0, y) : y < 0\} $

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(B) Let the centre of the circle be $(h, k)$ and its radius be $r$. Since the circle touches the $x$-axis,the radius $r = |k|$.Since the circle touches the circle ${x^2} + {(y - 1)^2} = 1$ (centre $(0, 1)$,radius $1$) externally,the distance between their centres is equal to the sum of their radii:$\sqrt {{{(h - 0)}^2} + {{(k - 1)}^2}} = 1 + |k|$Squaring both sides:${h^2} + {(k - 1)^2} = {(1 + |k|)^2}$${h^2} + {k^2} - 2k + 1 = 1 + 2|k| + {k^2}$${h^2} = 2k + 2|k|$Case $1$: If $k > 0$,then $|k| = k$,so ${h^2} = 2k + 2k = 4k$. Thus,the locus is ${x^2} = 4y$ for $y > 0$.Case $2$: If $k Combining these,the locus is $\{ (x, y) : {x^2} = 4y, y > 0\} \cup \{ (0, y) : y < 0\}$.

The locus of the centre of the circle which touches the circle ${x^2} + {(y - 1)^2} = 1$ externally and also touches the $x$-axis is

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