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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).
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Solution
Equation of the line passing through (acosα, asinα) and (acosβ, asinβ) is
\[y - asin\alpha = \frac{asin\beta - asin\alpha}{acos\beta - acos\alpha}\left( x - acos\alpha \right)\]
\[ \Rightarrow y - asin\alpha = \frac{sin\beta - sin\alpha}{cos\beta - cos\alpha}\left( x - acos\alpha \right)\]
\[ \Rightarrow y - asin\alpha = \frac{2\cos\left( \frac{\beta + \alpha}{2} \right)\sin\left( \frac{\beta - \alpha}{2} \right)}{2\sin\left( \frac{\beta + \alpha}{2} \right)\sin\left( \frac{\alpha - \beta}{2} \right)}\left( x - acos\alpha \right)\]
\[ \Rightarrow y - asin\alpha = - \cot\left( \frac{\beta + \alpha}{2} \right)\left( x - acos\alpha \right)\]
\[ \Rightarrow y - asin\alpha = - \cot\left( \frac{\alpha + \beta}{2} \right)\left( x - acos\alpha \right)\]
\[\Rightarrow x\cot\left( \frac{\alpha + \beta}{2} \right) + y - asin\alpha - acos\alpha \cot\left( \frac{\alpha + \beta}{2} \right) = 0\]
The distance of the line from the origin is
\[d = \left| \frac{- asin\alpha - acos\alpha \cot\left( \frac{\alpha + \beta}{2} \right)}{\sqrt{\cot^2 \left( \frac{\alpha + \beta}{2} \right) + 1}} \right|\]
\[ \Rightarrow d = \left| \frac{asin\alpha + acos\alpha \cot\left( \frac{\alpha + \beta}{2} \right)}{\sqrt{{cosec}^2 \left( \frac{\alpha + \beta}{2} \right)}} \right| \left( \because {cosec}^2 \theta = 1 + \cot^2 \theta \right)\]
\[\Rightarrow d = a\left| \sin\left( \frac{\alpha + \beta}{2} \right)sin\alpha + cos\alpha \cos\left( \frac{\alpha + \beta}{2} \right) \right| \]
\[ \Rightarrow d = a\left| sin\alpha \sin\left( \frac{\alpha + \beta}{2} \right) + cos\alpha \cos\left( \frac{\alpha + \beta}{2} \right) \right| \]
\[ \Rightarrow d = a\left| \cos\left( \frac{\alpha + \beta}{2} - \alpha \right) \right| = a\cos\left( \frac{\beta - \alpha}{2} \right) = a\cos\left( \frac{\alpha - \beta}{2} \right)\]
Hence, the required distance is \[a\cos\left( \frac{\alpha - \beta}{2} \right)\]
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