Question

Find the length of the perpendicular from the origin to the straight line joining the two points whose coordinates are (a cos α, a sin α) and (a cos β, a sin β).

Answer 1

Equation of the line passing through (acosα, asinα) and (acosβ, asinβ) is

$$\begin{gathered} y-a\mathrm{sin\alpha }=\frac{a\mathrm{sin\beta }-a\mathrm{sin\alpha }}{a\mathrm{cos\beta }-a\mathrm{cos\alpha }}\left(x-a\mathrm{cos\alpha }\right) \\ \Rightarrow y-a\mathrm{sin\alpha }=\frac{\mathrm{sin\beta }-\mathrm{sin\alpha }}{\mathrm{cos\beta }-\mathrm{cos\alpha }}\left(x-a\mathrm{cos\alpha }\right) \\ \Rightarrow y-a\mathrm{sin\alpha }=\frac{2\cos \left({\displaystyle \frac{\mathrm{\beta }+\mathrm{\alpha }}{2}}\right)\sin \left({\displaystyle \frac{\mathrm{\beta }-\mathrm{\alpha }}{2}}\right)}{2\sin \left({\displaystyle \frac{\mathrm{\beta }+\mathrm{\alpha }}{2}}\right)\sin \left({\displaystyle \frac{\mathrm{\alpha }-\mathrm{\beta }}{2}}\right)}\left(x-a\mathrm{cos\alpha }\right) \\ \Rightarrow y-a\mathrm{sin\alpha }=-\cot \left(\frac{\mathrm{\beta }+\mathrm{\alpha }}{2}\right)\left(x-a\mathrm{cos\alpha }\right) \\ \Rightarrow y-a\mathrm{sin\alpha }=-\cot \left(\frac{\mathrm{\alpha }+\mathrm{\beta }}{2}\right)\left(x-a\mathrm{cos\alpha }\right) \end{gathered}$$

$\Rightarrow x\cot \left(\frac{\mathrm{\alpha }+\mathrm{\beta }}{2}\right)+y-a\mathrm{sin\alpha }-a\mathrm{cos\alpha }\cot \left(\frac{\mathrm{\alpha }+\mathrm{\beta }}{2}\right)=0$

The distance of the line from the origin is

$$\begin{gathered} d=\left|\frac{-a\mathrm{sin\alpha }-a\mathrm{cos\alpha }\cot \left({\displaystyle \frac{\mathrm{\alpha }+\mathrm{\beta }}{2}}\right)}{\sqrt{{\cot }^2\left({\displaystyle \frac{\mathrm{\alpha }+\mathrm{\beta }}{2}}\right)+1}}\right| \\ \Rightarrow d=\left|\frac{a\mathrm{sin\alpha }+a\mathrm{cos\alpha }\cot \left({\displaystyle \frac{\mathrm{\alpha }+\mathrm{\beta }}{2}}\right)}{\sqrt{{\mathrm{cosec}}^2\left({\displaystyle \frac{\mathrm{\alpha }+\mathrm{\beta }}{2}}\right)}}\right|\left(\because {\mathrm{cosec}}^2\mathrm{\theta }=1+{\cot }^2\mathrm{\theta }\right) \end{gathered}$$

$$\begin{gathered} \Rightarrow d=a\left|\sin \left(\frac{\mathrm{\alpha }+\mathrm{\beta }}{2}\right)\mathrm{sin\alpha }+\mathrm{cos\alpha }\cos \left(\frac{\mathrm{\alpha }+\mathrm{\beta }}{2}\right)\right| \\ \Rightarrow d=a\left|\mathrm{sin\alpha }\sin \left(\frac{\mathrm{\alpha }+\mathrm{\beta }}{2}\right)+\mathrm{cos\alpha }\cos \left(\frac{\mathrm{\alpha }+\mathrm{\beta }}{2}\right)\right| \\ \Rightarrow d=a\left|\cos \left(\frac{\mathrm{\alpha }+\mathrm{\beta }}{2}-\mathrm{\alpha }\right)\right|=a\cos \left(\frac{\mathrm{\beta }-\mathrm{\alpha }}{2}\right)=a\cos \left(\frac{\mathrm{\alpha }-\mathrm{\beta }}{2}\right) \end{gathered}$$

Hence, the required distance is $a\cos \left(\frac{\mathrm{\alpha }-\mathrm{\beta }}{2}\right)$