Q. The lines x cos $\alpha$ + y sin  $\alpha$ = P_{1 & x cos $\beta$ + y sin}    $\beta$ = P₂ will be perpendicular if :
(a)  $\alpha$ $\pm$ $\beta$ = $\frac{\mathrm{\pi }}{2}$             (b)  $\alpha$ = $\frac{\mathrm{\pi }}{2}$                 (c) | $\alpha$- $\beta$| = $\frac{\mathrm{\pi }}{2}$               (d) $\alpha$ = $\beta$
 

Dear Student,
Please find below the solution to the asked query:

$$\begin{gathered} \mathrm{For} \mathrm{any} \mathrm{line} \mathrm{ax}+\mathrm{by}+\mathrm{c}=0 \\ \mathrm{Slope} \mathrm{of} \mathrm{line}=-\frac{\mathrm{Coefficient} \mathrm{of} \mathrm{x}}{\mathrm{Coefficient} \mathrm{of} \mathrm{y}} \\ \mathrm{Now} \\ \mathrm{xcos\alpha }+\mathrm{ysin\alpha }={\mathrm{p}}_1 \\ \mathrm{Slope}={\mathrm{m}}_1=-\frac{\mathrm{cos\alpha }}{\mathrm{sin\alpha }}=-\mathrm{cot\alpha } \\ \mathrm{xcos\beta }+\mathrm{ysin\beta }={\mathrm{p}}_2 \\ \mathrm{Slope}={\mathrm{m}}_2=-\frac{\mathrm{cos\beta }}{\mathrm{sin\beta }}=-\mathrm{cot\beta } \\ \mathrm{As} \mathrm{lines} \mathrm{are} \mathrm{perpendicular}, \mathrm{hence} \\ {\mathrm{m}}_1.{\mathrm{m}}_2=-1 \\ \Rightarrow \left(-\mathrm{cot\alpha }\right)\left(-\mathrm{cot\beta }\right)=-1 \\ \Rightarrow \mathrm{cot\alpha }.\mathrm{cot\beta }=-1 \\ \Rightarrow \mathrm{cot\alpha }=-\frac{1}{\mathrm{cot\beta }} \\ \Rightarrow \mathrm{cot\alpha }=-\mathrm{tan\beta } \\ \Rightarrow \mathrm{cot\alpha }=\cot \left(\frac{\mathrm{\pi }}{2}+\mathrm{\beta }\right) \mathrm{or} \mathrm{cot\alpha }=-\cot \left(\frac{\mathrm{\pi }}{2}-\mathrm{\beta }\right) \\ \Rightarrow \mathrm{cot\alpha }=\cot \left(\frac{\mathrm{\pi }}{2}+\mathrm{\beta }\right) \mathrm{or} \mathrm{cot\alpha }=\cot \left\{-\left(\frac{\mathrm{\pi }}{2}-\mathrm{\beta }\right)\right\} \\ \Rightarrow \mathrm{cot\alpha }=\cot \left(\frac{\mathrm{\pi }}{2}+\mathrm{\beta }\right) \mathrm{or} \mathrm{cot\alpha }=\cot \left\{\mathrm{\beta }-\frac{\mathrm{\pi }}{2}\right\} \\ \Rightarrow \mathrm{\alpha }=\frac{\mathrm{\pi }}{2}+\mathrm{\beta } \mathrm{or} \mathrm{a}=\mathrm{\beta }-\frac{\mathrm{\pi }}{2} \\ \Rightarrow \mathrm{\alpha }-\mathrm{\beta }=\frac{\mathrm{\pi }}{2} \mathrm{or} \mathrm{\alpha }-\mathrm{\beta }=-\frac{\mathrm{\pi }}{2} \\ \mathrm{Combining} \mathrm{answers} \mathrm{we} \mathrm{get} \left|\mathrm{\alpha }-\mathrm{\beta }\right|=\frac{\mathrm{\pi }}{2} \left[\mathrm{answer}\right] \end{gathered}$$

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