Q.):-  Find the equation of the chord of the circle $x^2+y^2=r^2$ passing through the point (2, 3) farthest from the centre.

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Please find below the solution to the asked query:

$$\begin{gathered} \mathrm{We} \mathrm{know} \mathrm{that} \mathrm{infinite} \mathrm{chord} \mathrm{pass} \mathrm{through} \mathrm{any} \mathrm{point} \mathrm{inside} \mathrm{circle} \\ \mathrm{out} \mathrm{of} \mathrm{which} \mathrm{farthest} \mathrm{from} \mathrm{centre} \mathrm{is} \mathrm{the} \mathrm{one} \mathrm{which} \mathrm{is} \mathrm{bisected} \mathrm{at} \mathrm{that} \mathrm{point}. \\ \mathrm{Hence} \left(2,3\right) \mathrm{will} \mathrm{be} \mathrm{mid} \mathrm{point} \mathrm{of} \mathrm{chord} \mathrm{and} \mathrm{hence} \mathrm{line} \mathrm{joining} \mathrm{centre}\left(0,0\right) \mathrm{and} \\ \left(2,3\right) \mathrm{will} \mathrm{be} \mathrm{perpendicular} \mathrm{to} \mathrm{chord}. \\ \mathrm{Slope} \mathrm{of} \mathrm{line} \mathrm{joining} \mathrm{centre} \mathrm{and} \left(2,3\right)={\mathrm{m}}_1=\frac{3-0}{2-0}=\frac{3}{2} \\ \mathrm{Hence} \mathrm{slope} \mathrm{of} \mathrm{chord}=\mathrm{m}=-\frac{1}{{\mathrm{m}}_1}=-\frac{2}{3} \\ \mathrm{Equation} \mathrm{of} \mathrm{chord} \mathrm{is} \\ \mathrm{y}-3=-\frac{2}{3}\left(\mathrm{x}-2\right) \\ \Rightarrow 3\mathrm{y}-9=-2\mathrm{x}+4 \\ \Rightarrow 2\mathrm{x}+3\mathrm{y}=13 \end{gathered}$$

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