Consider a family of circles which pass 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:
Solutions · 1
Official solution
Equation of circle whose center is \[\left( {h,k} \right)\]
i.e \[{\left( {x - h} \right)^2} + {\left( {y - k} \right)^2} = {k^2}\]
(radius of circle =k because circle is tangent to x-axis)
Equation of circle passing through \[\left( { - 1, + 1} \right)\]
\[\therefore\] \[{\left( { - 1, - h} \right)^2} + {\left( {1 - k} \right)^2} = {k^2}\]
\[ \Rightarrow 1 + {h^2} + 2h + 1 + {k^2} - 2k = {k^2}\]
\[ \Rightarrow {h^2} + 2h - 2k + 2 = 0\]
\[D \ge 0\]
\[\therefore\] \[{\left( 2 \right)^2} - 4 \times 1.\left( { - 2k + 2} \right) \ge 0\]
\[ \Rightarrow 4 - 4\left( { - 2k + 2} \right) \ge 0\]
\[ \Rightarrow 1 + 2k - 2 \ge 0\]
\[ \Rightarrow k \ge {1 \over 2}\]