a tangent is drawn to a cricle. 2(x^2+y^2)-3x+4y=0 and it touches the circle at point A. the tangent passes through the point P(2,1), then PA is equal to

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$$\begin{gathered} \mathrm{We} \mathrm{have}: \\ 2\left({\mathrm{x}}^2+{\mathrm{y}}^2\right)-3\mathrm{x}+4\mathrm{y}=0 \\ \mathrm{Divide} \mathrm{each} \mathrm{term} \mathrm{by} 2, \\ \Rightarrow {\mathrm{x}}^2+{\mathrm{y}}^2-\frac{3}{2}\mathrm{x}+2\mathrm{y}=0 \\ \mathbf{Note}\mathbf{:}\mathbf{ }\mathbf{If}\mathbf{ }\mathbf{tangent}\mathbf{ }\mathbf{is}\mathbf{ }\mathbf{drawn}\mathbf{ }\mathbf{to}\mathbf{ }\mathbf{cirlce}\mathbf{ }\mathbf{S}\mathbf{:}{\mathbf{x}}^{\mathbf{2}}\mathbf{+}{\mathbf{y}}^{\mathbf{2}}\mathbf{+}\mathbf{2}\mathbf{gx}\mathbf{+}\mathbf{2}\mathbf{fy}\mathbf{+}\mathbf{c}\mathbf{=}\mathbf{0}\mathbf{ }\mathbf{from}\mathbf{ }\mathbf{P}\left({\mathbf{x}}_{\mathbf{1}}\mathbf{,}{\mathbf{y}}_{\mathbf{1}}\right)\mathbf{ }\mathbf{which}\mathbf{ }\mathbf{touches} \\ \mathbf{the}\mathbf{ }\mathbf{circle}\mathbf{ }\mathbf{at}\mathbf{ }\mathbf{A}\mathbf{,}\mathbf{ }\mathbf{then}\mathbf{:} \\ \mathbf{Length}\mathbf{ }\mathbf{of}\mathbf{ }\mathbf{tangent}\mathbf{=}\mathbf{PA}\mathbf{=}\sqrt{{\mathbf{S}}_{\mathbf{1}}}\mathbf{=}\sqrt{{{\mathbf{x}}_{\mathbf{1}}}^{\mathbf{2}}\mathbf{+}{{\mathbf{y}}_{\mathbf{1}}}^{\mathbf{2}}\mathbf{+}\mathbf{2}{\mathbf{gx}}_{\mathbf{1}}\mathbf{+}\mathbf{2}{\mathbf{fy}}_{\mathbf{1}}\mathbf{+}\mathbf{c}} \\ \mathrm{We} \mathrm{have} \mathrm{P}\left(2,1\right) \mathrm{and} \mathrm{A}:{\mathrm{x}}^2+{\mathrm{y}}^2-\frac{3}{2}\mathrm{x}+2\mathrm{y}=0 \\ \Rightarrow \mathrm{PA}=\sqrt{{\left(2\right)}^2+{\left(1\right)}^2-\frac{3}{2}\left(2\right)+2\times 1}=\sqrt{4+1-3+2}=\sqrt{4}=2 \\ \mathrm{Hence} \mathrm{length} \mathrm{of} \mathrm{PA} \mathrm{is} 2 \mathrm{units}. \end{gathered}$$
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