Consider a family of circles which are passing through the point `(-1,1)` and are tangent to the x-axis. If `(h ,k)` are the coordinates of the center of the circles, then the set of values of `k` is given by the interval.
(a)`kgeq1/2` (b) `-1/2lt=klt=1/2` `klt=1/2` (d) `0`<`k`<`1/2`

Options

A. `(1)/(2) le k le (1)/(2)`
B. `k le (1)/(2)`
C. `0 lt k lt (1)/(2)`
D. `k ge (1)/(2)`

Answer

D

Explanation

The family of circles having (h,k) as the centre and touching x-axis is
`(x-h)^(2)+(y-k)^(2)=k^(2)`...(i)
It passes through (-1, 1)
`:. (h+1)^(2)+(1-k)^(2)=k^(2)`
`rArr h^(2)+2h+2-2k=0`
`rArr 4-4(2-2k)ge 0 rArr k ge (1)/(2) " " [:' "h is real" :. "Disc" gt 0]`