A circle is given by `x^2 + (y-1) ^2 = 1`, another circle C touches it externally and also the x-axis, then the locus of center is:

Options

A. `{(x,y):x^(2)=4y} uu {(0,y) : y lt 0}`
B. `{(x, y):y = x^(2)} uu {0, y):y le 0}`
C. `{(x, y) : x^(2)+(y-1)^(2)=4} uu {90, y) : y lt 0}`
D. `{(x,y) : x^(2)+4y=0} uu {(0, y) : y lt 0}`

Answer

A

Explanation

Let (h, k) be the centre of the circle and r be the radius. Then,
`sqrt((h-0)^(2)+(k-1)^(2))=r+1 and |K|=r`
`rArr h^(2)+(k-1)^(2)=(|k|+1)^(2)`
`rArr h^(2)=4k` when `k ge 0 or h = 0` when `k lt 0`
Hence, the locus of (h, k) is
`x^(2)=4y` when `y ge 0` or, x=0 when `ylt0`
or `{x^(2)=4y : y le 0} uu { (0, y) : y lt 0}`