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Q.3. Find the equations of the tangent and the normal to the curve $x^2+y^2=5$, where the tangent is parallel to the line 2x - y + 1 = 0

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Please find below the solution to the asked query:

$$\begin{gathered} \mathrm{We} \mathrm{know} \mathrm{that} \\ \mathrm{y}=\mathrm{mx}+\mathrm{c} \mathrm{is} \mathrm{tangent} \mathrm{to} \mathrm{circle} {\mathrm{x}}^2+{\mathrm{y}}^2={\mathrm{a}}^2 \mathrm{if} \\ {\mathrm{c}}^2={\mathrm{a}}^2\left(1+{\mathrm{m}}^2\right) \\ {\mathrm{x}}^2+{\mathrm{y}}^2=5 \\ {\mathrm{a}}^2=5 \\ \mathrm{As} \mathrm{tangent} \mathrm{is} \mathrm{parallel} \mathrm{to} 2\mathrm{x}-\mathrm{y}+1=0, \mathrm{hence} \\ \mathrm{m}=\mathrm{Slope} \mathrm{of} 2\mathrm{x}-\mathrm{y}+1=0 \\ \Rightarrow \mathrm{m}=2 \\ \mathrm{And} \mathrm{slope} \mathrm{of} \mathrm{normal}={\mathrm{m}}_{\mathrm{n}}=-\frac{1}{2} \\ \Rightarrow {\mathrm{c}}^2={\mathrm{a}}^2\left(1+{\mathrm{m}}^2\right)=5\left(1+2^2\right)=5^2 \\ \Rightarrow \mathrm{c}=\pm 5 \\ \mathrm{Hence} \mathrm{equation} \mathrm{of} \mathrm{tangents} \mathrm{are} \mathrm{y}=2\mathrm{x}\pm 5 \\ \mathrm{Now} \mathrm{normal} \mathrm{of} \mathrm{circle} \mathrm{passes} \mathrm{through} \mathrm{centre} \left(0,0\right) \mathrm{of} \mathrm{circle} \\ \mathrm{Hence} \mathrm{equation} \mathrm{is} \\ \mathrm{y}-0={\mathrm{m}}_{\mathrm{n}}\left(\mathrm{x}-0\right) \\ \Rightarrow \mathrm{y}=-\frac{1}{2}\mathrm{x} \\ \Rightarrow \mathrm{x}+2\mathrm{y}=0 \end{gathered}$$

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