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The Straight Lines — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHSThe Straight Lines3 Marks
Question
Find the perpendicular distance of the line joining the points $\big(\cos\theta, \sin \theta\big) $ and $\big(\cos \phi, \sin \phi\big)$ from the origin.
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Answer

Equation of line passing through $\big(\cos\theta, \sin \theta\big) $ and $\big(\cos \phi, \sin \phi\big)$ is $\text{y}-\sin\phi=\Big(\frac{\sin\phi-\sin\theta}{\cos\phi-\cos\theta}\Big)(\text{x}-\cos\phi)$ $\text{y}-\sin\phi=\Bigg(\frac{2\cos\frac{\theta+\phi}{2}\sin\frac{\phi-\theta}{2}}{-2\sin\frac {\theta+\phi} {2}\sin\frac{\phi-\theta}{2}}\Bigg)(\text{x}-\cos\phi)$ $\text{y}-\sin\phi=-\cot\Big(\frac{\theta+\phi}{2}\Big)(\text{x}-\cos\phi)$ $\text{x}\cot\Big(\frac{\theta+\phi}{2}\Big)+\text{y}-\sin\phi-\cos\phi\cot\Big(\frac{\theta+\phi}{2}\Big)=0$ Distance of this line from origin, $=\Big|\frac{\text{ax}_1+\text{by}_1+\text{c}}{\text{a}^2+\text{b}^2}\Big|$ $=\Bigg|\frac{0+0-\sin\phi-\cos\phi\cot\big(\frac{\theta+\phi}{2}\big)}{\sqrt{\Big(\cos\Big(\frac{\theta+\phi}{2}\Big)\Big)^2+1}}\Bigg|$ $=\Bigg|\frac{\sin\phi+\cos\phi\cot\big(\frac{\theta+\phi}{2}\big)}{{\text{cosec}\Big(\frac{\theta+\phi}{2}\Big)}}\Bigg|$ $=\sin\phi\sin\Big(\frac{\theta+\phi}{2}\Big)+{{\cos\phi\cos\Big(\frac{\theta+\phi}{2}}}\Big)$ $=\cos\Big(\frac{\theta+\phi}{2}-\phi\Big)$ $=\cos\Big(\frac{\theta+\phi-2\phi}{2}\Big)$ $\text{D}=\cos\Big(\frac{\theta-\phi}{2}\Big)$

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