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The locus of the centres of circles whic...

The locus of the centres of circles which touches `(y-1)^(2)+x^(2)=1` externally and also touches X-axis is

A

`{x^(2)=4y,yge0}uu{0,y),ylt}`

B

`x^(2)=y`

C

`y=4x^(2)`

D

`y^(2)=4x uu(0,y),yinR`

Text Solution

Verified by Experts

The correct Answer is:
A
Let the locus of centre of circle be (h, k) touching `(y-1)^(2)+x^(2)=1` = 1 and X-axis shown as

Clearly, from figure,
Distance between C and A is always 1 + `|k|`,
i.e. `sqrt((h-0)^(2)+(k-1)^(2)=1+|k|)`,
`rArrh^(2)+k^(2)-2k+1=1+k^(2)+2|k|`
`rArr h^(2)=2|k|+2krArrx^(2)=2|y|+2y`
where `|y| = {{:(y " , " y ge 0),(y " , " y lt0):}`
`therefore x^(2)=2y+2y, y ge 0`
`and x^(2)=-2y+2y, ylt0`
`rArr x^(2)=4y, " when "yge0`
`and x^(2)=0," when " y lt0`
`therefore {(x,y):x^(2)=4y, " when " yge0}uu{(0,y):ylt0}`
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