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 coordinate of the centre of the circles, then the set of values of $k$ is given by the interval

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

Answer 1

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

Equation of circle $(x-h{)}^2+(y-k{)}^2=k^2$

It is passing through $(-1,1)$ then

$(-1-h{)}^2+(1-k{)}^2=k^2\Rightarrow h^2+2h-2k+2=0$
For real value of $h$

$D\ge 0$ [Using $D=b^2-4ac$ for an equation $a^2+bx+c=0$]

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