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Question

If the line segment joining the points p(x1,y1) and Q(x2,y2)subtends and angle α at the origin O,prove that:OP.OQ cos α=x1 x2+y1 y2.

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Solution

It is given that O is the origin,Then,OQ2=x22+y22OP2=x21+y21and PQ2=(x2x1)2+(y2y1)2Using cosine formula in ΔOPQ,we havePQ2=OP2+OQ22(OP)(OQ)cos α(x2x1)2+(y2y1)2=x22+y22+x21+y212(OP),(OQ) cosαx22+x212x2x1+y22+y212y2y1=x22+y22+x21+y212OP.OQ cosα2x1x22y1y2=2OP.OQ cosαx1x2+y1y2=OP.OQ cosαOP.OQ cosα=x1x2+y1y2 Hence,proved.


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