If the limiting points of the system of circles x2+ y2+ 2gx +w( x2+ y2 + 2fy + k)=0 where w is a parameter , subtends a right angle at origin then k/f2?

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$$\begin{gathered} \mathrm{We} \mathrm{have}: \\ {\mathrm{x}}^2+{\mathrm{y}}^2+2\mathrm{gx}+\mathrm{w}\left({\mathrm{x}}^2+{\mathrm{y}}^2+2\mathrm{fy}+\mathrm{k}\right)=0 \\ \Rightarrow {\mathrm{x}}^2\left(1+\mathrm{w}\right)+{\mathrm{y}}^2\left(1+\mathrm{w}\right)+2\mathrm{gx}+2\mathrm{fyw}+\mathrm{wk}=0 \\ \mathrm{Divide} \mathrm{each} \mathrm{term} \mathrm{by} \left(1+\mathrm{w}\right) \\ \Rightarrow {\mathrm{x}}^2+{\mathrm{y}}^2+2\frac{\mathrm{g}}{1+\mathrm{w}}\mathrm{x}+2\frac{\mathrm{fw}}{1+\mathrm{w}}+\frac{\mathrm{wk}}{1+\mathrm{w}}=0 \\ \mathrm{By} \mathrm{method} \mathrm{of} \mathrm{completing} \mathrm{squares} \mathrm{we} \mathrm{get}: \\ {\left(\mathrm{x}+\frac{\mathrm{g}}{1+\mathrm{w}}\right)}^2-{\left(\frac{\mathrm{g}}{1+\mathrm{w}}\right)}^2+{\left(\mathrm{y}+\frac{\mathrm{fw}}{1+\mathrm{w}}\right)}^2-{\left(\frac{\mathrm{fw}}{1+\mathrm{w}}\right)}^2+\frac{\mathrm{kw}\left(1+\mathrm{w}\right)}{{\left(1+\mathrm{w}\right)}^2}=0 \\ \mathrm{After} \mathrm{rearranging} \mathrm{we} \mathrm{get}: \\ {\left(\mathrm{x}+\frac{\mathrm{g}}{1+\mathrm{w}}\right)}^2++{\left(\mathrm{y}+\frac{\mathrm{fw}}{1+\mathrm{w}}\right)}^2=\frac{{\mathrm{g}}^2}{{\left(1+\mathrm{w}\right)}^2}+\frac{{\mathrm{f}}^2{\mathrm{w}}^2}{{\left(1+\mathrm{w}\right)}^2}-\frac{\mathrm{kw}\left(1+\mathrm{w}\right)}{{\left(1+\mathrm{w}\right)}^2} \\ \Rightarrow {\left(\mathrm{x}+\frac{\mathrm{g}}{1+\mathrm{w}}\right)}^2++{\left(\mathrm{y}+\frac{\mathrm{fw}}{1+\mathrm{w}}\right)}^2=\frac{{\mathrm{g}}^2+{\mathrm{f}}^2{\mathrm{w}}^2-\mathrm{kw}\left(1+\mathrm{w}\right)}{{\left(1+\mathrm{w}\right)}^2} \\ \mathrm{This} \mathrm{is} \mathrm{nothing} \mathrm{but} \mathrm{a} \mathrm{circle} \mathrm{in} \mathrm{centre} \mathrm{radius} \mathrm{form}. \\ \mathrm{Hence} \mathrm{coordinates} \mathrm{of} \mathrm{centre} \mathrm{are}: \\ \left(-\frac{\mathrm{g}}{1+\mathrm{w}},-\frac{\mathrm{fw}}{1+\mathrm{w}}\right) \\ \mathrm{Also} \mathrm{for} \mathrm{limiting} \mathrm{case}, \mathrm{radius} \mathrm{of} \mathrm{circle} \mathrm{must} \mathrm{be} 0. \\ \Rightarrow \frac{{\mathrm{g}}^2+{\mathrm{f}}^2{\mathrm{w}}^2-\mathrm{kw}\left(1+\mathrm{w}\right)}{{\left(1+\mathrm{w}\right)}^2}=0 \\ \Rightarrow {\mathrm{g}}^2+{\mathrm{f}}^2{\mathrm{w}}^2-\mathrm{kw}\left(1+\mathrm{w}\right)=0 \\ \Rightarrow \left({\mathrm{f}}^2-\mathrm{k}\right){\mathrm{w}}^2-\mathrm{kw}+{\mathrm{g}}^2=0 \\ \mathrm{Above} \mathrm{is} \mathrm{a} \mathrm{quadratic} \mathrm{equation} \mathrm{in} \mathrm{w}. \\ \mathrm{Let} \mathrm{roots} \mathrm{be} {\mathrm{w}}_1,{\mathrm{w}}_2 \\ \mathrm{Hence} \mathrm{product} \mathrm{of} \mathrm{roots} \mathrm{will} \mathrm{be}: \\ {\mathrm{w}}_1{\mathrm{w}}_2=\frac{{\mathrm{g}}^2}{{\mathrm{f}}^2-\mathrm{k}} .....\left(1\right) \\ \mathrm{It} \mathrm{is} \mathrm{time} \mathrm{now} \mathrm{to} \mathrm{to} \mathrm{introduce} \mathrm{the} \mathrm{right} \mathrm{angle} \mathrm{condition} \mathrm{between} \overrightarrow{{\mathrm{OC}}_1} \mathrm{and} \overrightarrow{{\mathrm{OC}}_2}. \\ \mathrm{For} \mathrm{perpendicular} \mathrm{vectors}, \mathrm{dot} \mathrm{product} \mathrm{is} 0. \\ \Rightarrow \left(-\frac{\mathrm{g}}{1+{\mathrm{w}}_1}\right)\left(-\frac{\mathrm{g}}{1+{\mathrm{w}}_2}\right)+\left(-\frac{{\mathrm{fw}}_1}{1+{\mathrm{w}}_1}\right)-\frac{{\mathrm{fw}}_2}{1+{\mathrm{w}}_2}=0 \\ \mathrm{I} \mathrm{leave} \mathrm{simplification} \mathrm{of} \mathrm{this} \mathrm{equation} \mathrm{to} \mathrm{you} \mathrm{which} \mathrm{will} \mathrm{yield}. \\ {\mathrm{w}}_1{\mathrm{w}}_2=-\frac{{\mathrm{g}}^2}{{\mathrm{f}}^2} ...\left(2\right) \\ \mathrm{By} \left(1\right) \mathrm{and} \left(2\right) \\ \frac{{\mathrm{g}}^2}{{\mathrm{f}}^2-\mathrm{k}}=-\frac{{\mathrm{g}}^2}{{\mathrm{f}}^2} \\ \Rightarrow {\mathrm{f}}^2=-{\mathrm{f}}^2+\mathrm{k} \\ \Rightarrow 2{\mathrm{f}}^2=\mathrm{k} \\ \Rightarrow \frac{\mathbf{k}}{{\mathbf{f}}^{\mathbf{2}}}\mathbf{=}\mathbf{2} \\ \mathrm{I} \mathrm{would} \mathrm{like} \mathrm{to} \mathrm{suggest} \mathrm{you} \mathrm{that} \mathrm{this} \mathrm{topic} \mathrm{is} \mathrm{not} \mathrm{very} \mathrm{important} \mathrm{from} \mathrm{JEE} \mathrm{point} \mathrm{of} \mathrm{view}. \end{gathered}$$
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