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Question

The locus of the centre of circle which touches (y−1)2+x2=1 externally and also touches x-axis, is

A
{x2=4y,y0}{(0,y),y<0}
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B
x2=y
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C
y=4x2
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D
y2=4x(0,y),yϵR
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Solution

The correct option is A {x2=4y,y≥0}∪{(0,y),y<0}Let the locus of centre of circle be (h, k) touching (y−1)2+x2=1 and X- axis shown as Clearly, from figure, Distance between C and A is always 1+|k|, i.e.√(h−0)2+(k−1)2=1+|k| ⇒h2+k2−2k+1=1+k2+2|k|⇒h2=2|k|+2k⇒x2=2|y|+2y where |y|={y,y≥0−y,y<0 ∴ x2=2y+2y,y≥0 and x2=−2y+2y,y<0 ⇒x2=4y, when y≥0 and x2=0, when y<0 ∴{(x,y):x2=4y, when y≥0}∪{(0,y);y<0}

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