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 \[(a\,\cos \alpha ,a\sin \alpha )\] and \[(a\cos \beta ,a\sin \beta )\] is \[P\,\left( \frac{a(\cos \alpha +\cos \beta )}{2},\frac{a(\sin \alpha +\sin \beta )}{2} \right)\] Slope of line\[AB\]is \[\frac{a\sin \beta -a\sin \alpha }{a\cos \beta -a\cos \alpha }\]\[=\frac{\sin \beta -\sin \alpha }{\cos \beta -\cos \alpha }={{m}_{1}}\] and slope of \[OP\]is \[\frac{\sin \alpha +\sin \beta }{\cos \alpha +\cos \beta }={{m}_{2}}\] Now \[{{m}_{1}}\times {{m}_{2}}=\frac{{{\sin }^{2}}\beta -{{\sin }^{2}}\alpha }{{{\cos }^{2}}\beta -{{\cos }^{2}}\alpha }=-1\] Hence the lines are perpendicular.