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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 $r = |k|$. The circle also touche

Vedclass · Accepted Answer

$A$ parabola

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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 $r = |k|$. The circle also touches the circle with centre $(0, 3)$ and radius $2$. The distance between the centres is equal to the sum of the radii: $\sqrt{(h - 0)^2 + (k - 3)^2} = r + 2$. Substituting $r = |k|$,we get $\sqrt{h^2 + (k - 3)^2} = |k| + 2$. Squaring both sides: $h^2 + (k - 3)^2 = (|k| + 2)^2$. $h^2 + k^2 - 6k + 9 = k^2 + 4|k| + 4$. $h^2 = 4|k| + 6k + 5$. If $k > 0$,$h^2 = 10k + 5 = 10(k + 0.5)$,which represents a parabola. Thus,the locus of the centre $(x, y)$ is $x^2 = 10y + 5$.

$A$ circle touches the $x$-axis and also touches the circle with centre at $(0, 3)$ and radius $2$. The locus of the centre of the circle is

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