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