question_answer

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)               \[k\ge \frac{1}{2}\]

Correct Answer: D

Solution :

Equation of circle whose centre is (h, k) is given as \[{{(x-h)}^{2}}+{{(y-k)}^{2}}={{k}^{2}}\]

(radius of circle = k because circle is tangent to x-axis)

Equation of circle passing through \[(-1,+1)\]

\[\therefore \,\,{{(-1-h)}^{2}}+{{(1-k)}^{2}}={{k}^{2}}\]

\[\Rightarrow \,\,1+{{h}^{2}}+2h+1+{{k}^{2}}-2k={{k}^{2}}\]

\[\Rightarrow \,\,{{h}^{2}}+2h-2k+2=0\]

Since, it is a quadratic in 'h' and \[D\ge 0\]

\[\therefore \,\,{{(2)}^{2}}-4\times 1.(-2k+2)\ge 0\]

\[\Rightarrow \,\,4-4(-2k+2)\ge 0\Rightarrow 1+2k-2\ge 0\Rightarrow k\ge \frac{1}{2}\]