Question

Consider a family of circles which are passing through the point (-1, 1) and are tangent to 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 (1,0)

• A. None
• B. $k\ge \frac{1}{2}$
• C. $-\frac{1}{2}\le k\le \frac{1}{2}$
• D. $k\le \frac{1}{2}$

Answer 1

The correct option is A None

We have

Circle passing through the point $(-1,1)$

If $(h,k)$ are the coordinates of the centre of the circles,

Now,

According to given question

Centre of circle is $=(h,k)$.

It touches X-axis its radius is k,

So,

$\begin{align}

${(x-h)}^2+{(y-k)}^2=R^2$

$\Rightarrow {(x-h)}^2+{(y-k)}^2=k^2(\therefore Radius=k)$

As the points $(-1,1)$.

So,

$h^2+2h-2k+2=0$

Here,

Discriminate is greater than or equal to zero for real values of h.

So,

$2k-1\ge 0$

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

So, the set of values of $k\in [\frac{1}{2},\infty )$

Hence, this is the answer.