Question

Consider a family of circles which are passing through the point (-1, 1) and are tangent to the x-axis. If (h, k) are the coordinates 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. $-\frac{1}{2}\le k\le \frac{1}{2}$
• D. $k\le \frac{1}{2}$

Answer 1

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

The equation of the circle is
$(x-h{)}^2+(y-k{)}^2=k^2(\because circletouchesx-axis)$
It passes through (-1, 1), then
$(-1-h{)}^2+(1-k{)}^2=k^2\Rightarrow h^2+2h-2k+2=0Sincehisreal\Rightarrow D\ge 0\Rightarrow 4-4(-2k+2)\ge 0\Rightarrow 2k-1\ge 0\Rightarrow k\ge \frac{1}{2}$