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(B) Let the circle have center $(h, k)$ and radius $r$.Since the circle touches the lines $y = a$ and $y = b$,the diameter of the circle must be equal

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$2$

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(B) Let the circle have center $(h, k)$ and radius $r$.Since the circle touches the lines $y = a$ and $y = b$,the diameter of the circle must be equal to the distance between these two parallel lines.Thus,$2r = |b - a|$,which implies $r = \frac{|b - a|}{2}$.The distance from the center $(h, k)$ to the line $y = a$ is $|k - a| = r$,so $k = a \pm r$.The distance from the center $(h, k)$ to the line $x = 0$ is $|h| = r$,so $h = \pm r$.Since $r$ is fixed,for each value of $h$ ($r$ or $-r$),we get a circle. Thus,there are exactly $2$ such circles.

The number of circles touching the lines $x = 0$,$y = a$ and $y = b$ is

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