question_answer

A straight line through origin bisect the line passing through the given points \[(a\,cos\,\alpha ,\,\,a\,sin\alpha )\] and \[(a\,\,\cos \beta ,a\sin \beta ),\]then the lines are

A) Perpendicular

B) Parallel

C) Angle between them is \[\frac{\pi }{4}\]

D) None of these

Correct Answer: A

Solution :

Mid point of the points \[(a\cos \alpha ,a\sin \alpha )\] and \[(a\cos \beta ,a\sin \beta )\] is, \[P\left( \frac{a(\cos \alpha +\cos \beta }{2},\frac{a(\sin \alpha +\sin \beta )}{2} \right)\]. Now, slope of line AB is \[\frac{a\sin \beta -a\sin \alpha }{a\cos \beta -a\cos \alpha }=\frac{\sin \beta -\sin \alpha }{\cos \beta -\cos \alpha }={{m}_{1}}\]      ?.(i) and slope of OP is \[\frac{\sin \alpha +\sin \beta }{\cos \alpha +\cos \beta }={{m}_{2}}\]         ...(ii) Multiplying equations (i) and (ii), we get \[{{m}_{1}}\times {{m}_{2}}=\frac{{{\sin }^{2}}\beta -{{\sin }^{2}}\alpha }{{{\cos }^{2}}\beta -{{\cos }^{2}}\alpha }=-1\] Hence, the lines are perpendicular.