Question

The distance between the points $ (a\cos \alpha ,a\sin \alpha ) $ and $ a\cos \beta ,a\sin \beta $ is

• A. $ 2\left| \sin \left( \frac{\alpha -\beta }{2} \right) \right| $
• B. $ 2\left| a\sin \left( \frac{\alpha -\beta }{2} \right) \right| $
• C. $ 2\left| a\cos \left( \frac{\alpha -\beta }{2} \right) \right| $
• D. $ \left| a\cos \left( \frac{\alpha -\beta }{2} \right) \right| $
• E. $ 2|a(1-\cos (\alpha -\beta ))| $

Answer 1

Answer: (B)

Points $ (a\cos \alpha ,a\sin \alpha ) $ and $ (a\cos \beta ,a\sin \beta ) $ , distance between the points
$=\sqrt{{{(a\cos \alpha -a\cos \beta )}^{2}}+{{(a\sin \alpha -a\sin \beta )}^{2}}} $
$=\sqrt{\begin{align} & {{a}^{2}}({{\cos }^{2}}\alpha +{{\cos }^{2}}\beta +{{\sin }^{2}}\alpha +{{\sin }^{2}}\beta \\ & \,\,\,\,\,\,\,\,\,\,\,-2\cos \alpha .\cos \beta +2\sin \alpha .\sin \beta ) \\ \end{align}} $
$=|a|\sqrt{\begin{align} & ({{\sin }^{2}}\alpha +{{\cos }^{2}}\alpha )+({{\sin }^{2}}\beta +{{\cos }^{2}}\beta ) \\ & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,-2(\cos \alpha .\cos \beta +\sin \alpha .\sin \beta ) \\ \end{align}} $
$=|a|\sqrt{2-2\cos (\alpha -\beta )} $
$=|a|\sqrt{2}\sqrt{1-\cos (\alpha -\beta )} $
$=\sqrt{2}|a|\sqrt{2{{\sin }^{2}}\frac{(\alpha -\beta )}{2}} $
$=|a|\,\sqrt{2}\,\sqrt{2}\,\left| \sin \,\frac{(\alpha -\beta )}{2} \right| $
$=2\left| a.\sin \frac{(\alpha -\beta )}{2} \right| $