Question

Consider a family of circles which are passing through M(1,1) and are tangent to x-axis. If (h,k) be the center of the circle the
a) k > .5 c) 0 < k < .5
b) k < .5 d) -.5 < k < .5

Answer 1

Dear student
$$\begin{gathered} \mathrm{Circle}\mathrm{touching}\mathrm{x}-\mathrm{axis}\mathrm{and}\mathrm{having}(\mathrm{h},\mathrm{k})\mathrm{as}\mathrm{centre} \\ (\mathrm{x}-\mathrm{h}{)}^2+(\mathrm{y}-\mathrm{k}{)}^2={\mathrm{k}}^2 \\ \mathrm{Passing}\mathrm{through}(1,1);(1-\mathrm{h}{)}^2+(1-\mathrm{k}{)}^2={\mathrm{k}}^2 \\ \mathrm{Thus}\mathrm{we}\mathrm{get}:{\mathrm{h}}^2-2\mathrm{h}-2\mathrm{k}+2=0 \\ \mathrm{Since},\mathrm{D}\ge 0\Rightarrow {\left(-2\right)}^2-4\left(-2\mathrm{k}+2\right)=2\mathrm{k}-1\ge 0 \\ \Rightarrow \mathrm{k}\ge \frac{1}{2} \end{gathered}$$
Regards