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. $-\frac{1}{2}\le k\le \frac{1}{\begin{array}{c}2\\ \end{array}}$
• D. $k\le \frac{1}{2}$

Answer 1

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

ATQ the circle has centre $=(h,k)$

as it touches x axis,its radius is $k$, so its eqn would be:

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

$(-1,1)\Rightarrow (-1-h{)}^2+(1-k{)}^2=k^2$

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

here discriminant is greater than or equal to $0$ for real values of $h$

so, $2k-1\ge 0$

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