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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

• Option 1)

$x=\sqrt{1+4y},y\geq 0$

• Option 2)

$y=\sqrt{1+2x},x\geq 0$

• Option 3)

$y=\sqrt{1+4x},x\geq 0$

• Option 4)

$x=\sqrt{1+2y},y\geq 0$

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Let the centre of one such circle be P(h,k) since it touches y-axis in the first quadrant.

Now, since it touches $x^{2}+y^{2}=1$

$h+1=\sqrt{h^{2}+k^{2}}$

=> $h^{2}+2h+1=h^{2}+k^{2}$

=> $2h+1=k^{2}$

Hence the required locus is

$2x+1=y^{2}$

$\sqrt{2x+1}=y; x\geq 0$

So, option (2) is correct

Option 1)

$x=\sqrt{1+4y},y\geq 0$

Option 2)

$y=\sqrt{1+2x},x\geq 0$

Option 3)

$y=\sqrt{1+4x},x\geq 0$

Option 4)

$x=\sqrt{1+2y},y\geq 0$

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