Find the equation of circle with radius 2 and touching x-axis , such that the centre is in 3rdquadrant, and centre lies on y = 3x

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$$\begin{gathered} \mathrm{Let} \mathrm{equation} \mathrm{of} \mathrm{cirlce} \mathrm{be} \\ {\mathrm{x}}^2+{\mathrm{y}}^2+2\mathrm{gx}+2\mathrm{fy}+\mathrm{c}=0 \\ \mathrm{Now} \\ \mathrm{r}=2=\sqrt{{\mathrm{g}}^2+{\mathrm{f}}^2-\mathrm{c}} \\ \Rightarrow {\mathrm{g}}^2+{\mathrm{f}}^2-\mathrm{c}=4....\left(\mathrm{i}\right) \\ \mathrm{Centre} \left(-\mathrm{g},-\mathrm{f}\right) \mathrm{lies} \mathrm{on} \mathrm{y}=3\mathrm{x} \\ \Rightarrow \mathrm{f}=3\mathrm{g}....\left(\mathrm{ii}\right) \\ \mathrm{Also} \mathrm{circle} \mathrm{touches} \mathrm{x}-\mathrm{axis} \\ \Rightarrow \mathrm{X} \mathrm{intercept} \mathrm{in} 0 \\ \Rightarrow 2\sqrt{{\mathrm{g}}^2-\mathrm{c}}=0 \\ \Rightarrow {\mathrm{g}}^2=\mathrm{c}....\left(\mathrm{iii}\right) \\ \mathrm{Put} \mathrm{in} \left(\mathrm{i}\right) \\ \mathrm{c}+{\mathrm{f}}^2-\mathrm{c}=4 \\ \Rightarrow \mathrm{f}=\pm 2 \\ \mathrm{But} \mathrm{centre} \mathrm{lies} \mathrm{in} \mathrm{third} \mathrm{quadrant}, \mathrm{hence} \\ \mathrm{f}=2 \left(-\mathrm{f}=-2\right) \\ \mathrm{Put} \mathrm{in} \left(\mathrm{ii}\right) \\ 2=3\mathrm{g} \\ \Rightarrow \mathrm{g}=\frac{3}{2} \\ \mathrm{c}={\mathrm{g}}^2=\frac{9}{4} \\ \mathrm{We} \mathrm{have} \mathrm{g},\mathrm{f},\mathrm{c}. \mathrm{Put} \mathrm{in} \mathrm{equation} \mathrm{of} \mathrm{circle} \mathrm{and} \mathrm{obtain} \mathrm{the} \mathrm{equation}. \end{gathered}$$

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