The line lx + my + n = 0 intersects the cervue ax² + 2hxy + by² = 1. at points P and Q which lie at finite distances from the origin. If The circle on PQ as diameter passes through the origin then l² + m² will be ?

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$$\begin{gathered} \mathrm{This} \mathrm{is} \mathrm{question} \mathrm{of} \mathrm{Homegenization} \\ \mathrm{As} \mathrm{circle} \mathrm{with} \mathrm{PQ} \mathrm{as} \mathrm{diameter} \mathrm{passes} \mathrm{through} \mathrm{origin} \mathrm{O}, \mathrm{hence} \mathrm{angle} \\ \angle \mathrm{POQ}=90° \left[\mathrm{Angle} \mathrm{in} \mathrm{semi} \mathrm{circle} \mathrm{is} 90°\right] \\ \mathrm{Hence} \mathrm{on} \mathrm{Homegenization} \mathrm{of} \mathrm{line} \mathrm{and} \mathrm{curve}, \mathrm{we} \mathrm{must} \mathrm{get} \mathrm{perpendicular} \\ \mathrm{pair} \mathrm{of} \mathrm{lines} \\ {\mathrm{ax}}^2+2\mathrm{hxy}+{\mathrm{by}}^2=1 \\ \Rightarrow {\mathrm{ax}}^2+2\mathrm{hxy}+{\mathrm{by}}^2-1=0 \\ \Rightarrow {\mathrm{ax}}^2+2\mathrm{hxy}+{\mathrm{by}}^2-1^2=0...\left(\mathrm{i}\right) \\ \mathrm{lx}+\mathrm{my}+\mathrm{n}=0 \\ \Rightarrow \mathrm{lx}+\mathrm{my}=-\mathrm{n} \\ \Rightarrow -\left(\frac{\mathrm{lx}+\mathrm{my}}{\mathrm{n}}\right)=1 \\ \mathrm{Put} \mathrm{value} \mathrm{of} 1 \mathrm{in} \left(\mathrm{i}\right) \\ \Rightarrow {\mathrm{ax}}^2+2\mathrm{hxy}+{\mathrm{by}}^2-{\left(-\left(\frac{\mathrm{lx}+\mathrm{my}}{\mathrm{n}}\right)\right)}^2=0 \\ \Rightarrow {\mathrm{ax}}^2+2\mathrm{hxy}+{\mathrm{by}}^2-\frac{{\mathrm{l}}^2{\mathrm{x}}^2+{\mathrm{m}}^2{\mathrm{y}}^2+2\mathrm{mlxy}}{{\mathrm{n}}^2}=0 \\ \Rightarrow {\mathrm{an}}^2{\mathrm{x}}^2+2{\mathrm{hn}}^2\mathrm{xy}+{\mathrm{bn}}^2{\mathrm{y}}^2-{\mathrm{l}}^2{\mathrm{x}}^2-{\mathrm{m}}^2{\mathrm{y}}^2-2\mathrm{mlxy}=0 \\ \Rightarrow {\mathrm{x}}^2\left({\mathrm{an}}^2-{\mathrm{l}}^2\right)+\mathrm{xy}\left(2{\mathrm{hn}}^2-2\mathrm{ml}\right)+{\mathrm{y}}^2\left({\mathrm{bn}}^2-{\mathrm{m}}^2\right)=0 \\ \mathrm{For} \mathrm{perpendicular} \mathrm{of} \mathrm{lines} \\ \mathrm{Coefficient} \mathrm{of} {\mathrm{x}}^2+\mathrm{Coefficient} \mathrm{of} {\mathrm{y}}^2=0 \\ \Rightarrow {\mathrm{an}}^2-{\mathrm{l}}^2+{\mathrm{bn}}^2-{\mathrm{m}}^2=0 \\ \Rightarrow {\mathrm{l}}^2+{\mathrm{m}}^2={\mathrm{an}}^2+{\mathrm{bn}}^2 \\ \mathbf{\Rightarrow }{\mathbf{l}}^{\mathbf{2}}\mathbf{+}{\mathbf{m}}^{\mathbf{2}}\mathbf{=}{\mathbf{n}}^{\mathbf{2}}\left(\mathbf{a}\mathbf{+}\mathbf{b}\right) \end{gathered}$$

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