Find the equation of the line passing through the point (11,-2) and touching the circle x²+y²=25

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$$\begin{gathered} \mathrm{We} \mathrm{have} \\ {\mathrm{x}}^2+{\mathrm{y}}^2=25 \\ \Rightarrow {\left(\mathrm{x}-0\right)}^2+{\left(\mathrm{y}-0\right)}^2=5^2 \\ \mathrm{On} \mathrm{comparing} \mathrm{it} \mathrm{with}: \\ {\left(\mathrm{x}-\mathrm{h}\right)}^2+{\left(\mathrm{y}-\mathrm{k}\right)}^2={\mathrm{r}}^2 \\ \mathrm{We} \mathrm{get} \mathrm{Centre}\left(\mathrm{h},\mathrm{k}\right)=\mathrm{C}\left(0,0\right) \\ \mathrm{Radius}\left(\mathrm{r}\right)=5 \mathrm{units} \\ \mathrm{Let} \mathrm{required} \mathrm{equation} \mathrm{of} \mathrm{line} \mathrm{be}: \\ \mathrm{mx}-\mathrm{y}+\mathrm{c}=0 \\ \mathrm{Line} \mathrm{passes} \mathrm{through} \left(11,-2\right) \\ 11\mathrm{m}-\left(-2\right)+\mathrm{c}=0 \\ \Rightarrow 11\mathrm{m}+2+\mathrm{c}=0 \\ \Rightarrow \mathrm{c}=-11\mathrm{m}-2 \\ \Rightarrow \mathrm{mx}-\mathrm{y}+\mathrm{c}=0 \mathrm{becomes} \\ \mathrm{mx}-\mathrm{y}-11\mathrm{m}-2=0 \\ \mathrm{As} \mathrm{line} \mathrm{touches} \mathrm{circle} \mathrm{hence} \mathrm{length} \mathrm{of} \mathrm{perpendicular} \mathrm{from} \mathrm{center} \mathrm{of} \mathrm{circle} \\ \mathrm{to} \mathrm{line} \mathrm{will} \mathrm{be} \mathrm{equal} \mathrm{to} \mathrm{radius} \mathrm{of} \mathrm{circle}. \\ \frac{\left|0-0-11\mathrm{m}-2\right|}{\sqrt{{\mathrm{m}}^2+{\left(-1\right)}^2}}=5 \\ \Rightarrow \frac{\left|-\left(11\mathrm{m}+2\right)\right|}{\sqrt{{\mathrm{m}}^2+1}}=5 \\ \mathrm{Squaring} \mathrm{both} \mathrm{sides} \mathrm{we} \mathrm{get}, \\ \Rightarrow \frac{{\left(11\mathrm{m}+2\right)}^2}{1+{\mathrm{m}}^2}=25 \\ \Rightarrow 121{\mathrm{m}}^2+4+44\mathrm{m}=25+25{\mathrm{m}}^2 \\ \Rightarrow 121{\mathrm{m}}^2-25{\mathrm{m}}^2+44\mathrm{m}+4-25=0 \\ \Rightarrow 96{\mathrm{m}}^2+44\mathrm{m}-21=0 \\ \Rightarrow 96{\mathrm{m}}^2+72\mathrm{m}-28\mathrm{m}-21=0 \\ \Rightarrow 24\mathrm{m}\left(4\mathrm{m}+3\right)-7\left(4\mathrm{m}+3\right)=0 \\ \Rightarrow \left(24\mathrm{m}-7\right)\left(4\mathrm{m}+3\right)=0 \\ \Rightarrow \mathrm{m}=\frac{7}{24} \mathrm{or} \mathrm{m}=-\frac{3}{4} \\ \mathrm{We} \mathrm{have} \mathrm{mx}-\mathrm{y}-11\mathrm{m}-2=0 \\ \mathrm{When} \mathrm{m}=\frac{7}{24} \\ \Rightarrow \frac{7\mathrm{x}}{24}-\mathrm{y}-\frac{77}{24}-2=0 \\ \Rightarrow \frac{7\mathrm{x}-24\mathrm{y}-77-48}{24}=0 \\ \Rightarrow \mathbf{7}\mathbf{x}\mathbf{-}\mathbf{24}\mathbf{y}\mathbf{-}\mathbf{125}\mathbf{=}\mathbf{0} \\ \mathrm{When} \mathrm{m}=-\frac{3}{4} \\ -\frac{3\mathrm{x}}{4}-\mathrm{y}+\frac{33}{4}-2=0 \\ \mathrm{Multiply} \mathrm{both} \mathrm{sides} \mathrm{by} -4, \\ 3\mathrm{x}+4\mathrm{y}-33+8=0 \\ \Rightarrow \mathbf{3}\mathbf{x}\mathbf{+}\mathbf{4}\mathbf{y}\mathbf{-}\mathbf{25}\mathbf{=}\mathbf{0} \end{gathered}$$

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