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 co-ordinates of the centre of the circles, then the set of values of k is given by the interval

• A. $0<k<\frac{1}{2}$
• B. $k\ge \frac{1}{2}$
• C. $\frac{1}{2}>k>\frac{1}{2}$
• D. $k$

Answer 1

The correct option is B $k\ge \frac{1}{2}$

Equation of circle which touches x-axis and coordinate of center are $(h,k)$ is

${(x-h)}^2+{(y-k)}^2=k^2$
Since it is passing through $(-1,1)$, then

${(-1-h)}^2+{(1-k)}^2=k^2\Rightarrow h^2+2h-2k+2=0\because D\ge 0$

$\Rightarrow 2k-1\ge 0\Rightarrow k\ge \frac{1}{2}$