Let ax + by +c=0 be variable st. line , where a,b,c are 1st , 2^nd, 7^th terms of some increasing A.P .Then the variable st. line alwys passes through a fixed point which lies on
1. x²+ y²=13
2.x²+y²=5
3.y²=9/2x
4. 3x+4y=9

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$$\begin{gathered} \mathrm{We} \mathrm{have}: \\ \mathrm{ax}+\mathrm{by}+\mathrm{c}=0 .............\left(\mathrm{i}\right) \\ \mathrm{Let} \mathrm{A} \mathrm{be} \mathrm{first} \mathrm{term} \mathrm{of} \mathrm{A}.\mathrm{P}. \mathrm{and} \mathrm{D} \mathrm{be} \mathrm{the} \mathrm{common} \mathrm{difference}. \\ \mathrm{Given} \mathrm{that} \mathrm{a},\mathrm{b},\mathrm{c} \mathrm{are} 1\mathrm{st}, 2\mathrm{nd}, 7\mathrm{th} \mathrm{term} \mathrm{of} \mathrm{A}.\mathrm{P}., \mathrm{hence} \\ \mathrm{a}=\mathrm{A} \\ \mathrm{b}=\mathrm{A}+\mathrm{D} \\ \mathrm{c}=\mathrm{A}+6\mathrm{D} \\ \mathrm{Hence} \left(\mathrm{i}\right) \mathrm{becomes}: \\ \mathrm{Ax}+\left(\mathrm{A}+\mathrm{D}\right)\mathrm{y}+\mathrm{A}+6\mathrm{D}=0 \\ \Rightarrow \mathrm{Ax}+\mathrm{Ay}+\mathrm{Dy}+\mathrm{A}+6\mathrm{D}=0 \\ \Rightarrow \mathrm{A}\left(\mathrm{x}+\mathrm{y}+1\right)+\mathrm{D}\left(\mathrm{y}+6\right)=0 \\ \mathrm{This} \mathrm{is} \mathrm{equation} \mathrm{of} \mathrm{family} \mathrm{of} \mathrm{lines} \mathrm{passing} \mathrm{through} \mathrm{intersection} \mathrm{of} \\ \mathrm{x}+\mathrm{y}+1=0 ............\left(\mathrm{ii}\right) \\ \mathrm{y}+6=0 \\ \Rightarrow \mathrm{y}=-6 \\ \mathrm{Put} \mathrm{this} \mathrm{in} \left(\mathrm{ii}\right) \\ \Rightarrow \mathrm{x}-6+1=0 \\ \Rightarrow \mathrm{x}=5 \\ \mathrm{Hence} \left(\mathrm{x},\mathrm{y}\right)=\left(5,6\right) \\ \mathrm{But} \left(\mathrm{x},\mathrm{y}\right)=\left(5,6\right) \mathrm{does} \mathrm{not} \mathrm{lie} \mathrm{on} \mathrm{any} \mathrm{of} \mathrm{given} \mathrm{options}. \\ \mathrm{Hence} \mathrm{you} \mathrm{should} \mathrm{recheck} \mathrm{your} \mathrm{question} \mathrm{because} \mathrm{process} \mathrm{followed} \mathrm{is} \mathrm{correct}. \end{gathered}$$
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