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Q.

Consider a family of circles which passes through the point $(- 1, 1)$ and are tangents to $x$ -axis. If $(h, k)$ is the centre of a circle, then

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a

$k \geq \frac{1}{2}$

b

$- \frac{1}{2} \leq k \leq \frac{1}{2}$

c

$k \leq \frac{1}{2}$

d

$0 < k < \frac{1}{2}$

answer is A.

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Detailed Solution

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Given, centre $= (h, k),$ radius $= k$
The circles is $(x - h)^{2} + (y - k)^{2} = k^{2} .$
The point $(- 1, 1)$ lies on it.
$\begin{aligned} (- 1 - h)^{2} + (1 - k)^{2} = k^{2} \\ \Rightarrow k = \frac{(1 + h)^{2} + 1}{2} \geq \frac{1}{2} . \end{aligned}$

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