Question

THE STRIGHT LINE CUTS OFF INTERCEPTS FROM THE COORDINTE AXES ,THE SUM OF RECIPROCALS OF WHICH IS CONSTANT .SHOW THAT IT ALEAYS PASSES THRUA FIXED POINT

Answer 1

$$\begin{gathered} \mathrm{Let}\mathrm{the}\mathrm{equation}\mathrm{of}\mathrm{line}\mathrm{be}\frac{\mathrm{x}}{\mathrm{a}}+\frac{\mathrm{y}}{\mathrm{b}}=1...\left(\mathrm{i}\right) \\ \mathrm{i},\mathrm{e},\mathrm{Its}\mathrm{intercepts}\mathrm{on}\mathrm{the}\mathrm{axes}\mathrm{are}\mathrm{a}\mathrm{and}\mathrm{b}\mathrm{respectively}. \\ \mathrm{Now}, \\ \mathrm{According}\mathrm{to}\mathrm{question}, \\ \frac{1}{\mathrm{a}}+\frac{1}{\mathrm{b}}=\mathrm{constant}=\frac{1}{\mathrm{p}}\left(\mathrm{say}\right) \\ \Rightarrow \frac{1}{\mathrm{a}}+\frac{1}{\mathrm{b}}=\frac{1}{\mathrm{p}} \\ \Rightarrow \frac{\mathrm{p}}{\mathrm{a}}+\frac{\mathrm{p}}{\mathrm{b}}=1....\left(\mathrm{ii}\right) \\ \mathrm{From}\left(\mathrm{ii}\right),\mathrm{it}\mathrm{follows}\mathrm{that}\mathrm{the}\mathrm{line}\left(\mathrm{i}\right)\mathrm{passes}\mathrm{through}\mathrm{the}\mathrm{fixed}\mathrm{point}(\mathrm{p},\mathrm{p}). \end{gathered}$$