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The locus of the centres of the circles,...

The locus of the centres of the circles, which touch the circle, ` x^(2)+y^(2)=1` externally, also touch the Y-axis and lie in the first quadrant, is

A

`y=sqrt(1+2x,)xge0`

B

`y=sqrt(1+4x,)xge0`

C

`y=sqrt(1+2y,)yge0`

D

`y=sqrt(1+4y,)yge0`

Text Solution

Verified by Experts

The correct Answer is:
A
Let (h,k) be the centre of the circle and radius r = h as circle touch the Y-axis and other circle `x^(2)+y^(2) = 1` whose centre (0,0) and radius is 1.

`therefore OC = r + 1`
[`therefore` if circles touch each other externally, then `C_(1)C_(2)=r_(1)+r_(2)`]
`rArr sqrt(h^(2)+k^(2))=h+1, hgt0`
and `kgt0`, for first quadrant.
`rArr h^(2)+k^(2)=h^(2)+2h+1`
`rArr k^(2)=2h+1`
`rArrk=sqrt(1+2h), as kgt0`
Now, on taking locus of centre (h,k),we get
`y = sqrt(1+2x),xge0`
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Knowledge Check

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

    A
    `{(x, y):x^(2)+(y-1)^(2)=4} cup {(x,y): y lt 0}`
    B
    `{(x,y):x^(2)=4y} cup {(0, y) : y lt 0}`
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    D
    `{(x,y):x^(2)=4y} cup {(x,y) : y gt 0}`

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

    A
    `{(x, y):x^(2)+(y-1)^(2)=4} cup {(x,y): y gt 0}`
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    `{(x,y):x^(2)=4y} cup {(0, y) : (y lt 0)}`
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    `{(x,y):x^(2)=y}cup {(0,y): y lt 0}`
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  • The locus of the centres of circles which touches (y-1)^(2)+x^(2)=1 externally and also touches X-axis is

    A
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    B
    `x^(2)=y`
    C
    `y=4x^(2)`
    D
    `y^(2)=4x uu(0,y),yinR`

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