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Question

Let two non-collinear unit vectors ^a and ^b form an acute angle. A point P moves so that at any time t the position vector
OP (where, O is the origin) is given by
^acos t+^bsin t. When P is farthest from origin O, let M be the length of OP and ^u be the unit vector along OP. Then,


A

^u=^a+^b|^a+^b| and M=(1+^a.^b)12

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B

^u=^a^b|^a^b| and M=(1+^a.^b)12

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C

^u=^a+^b|^a+^b| and M=(1+2^a.^b)12

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D

^u=^a^b|^a^b| and M=(1+2^a.^b)12

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Solution

The correct option is A

^u=^a+^b|^a+^b| and M=(1+^a.^b)12


Given, OP=^a cos t+^b sin t|OP|=(^a.^a) cos2 t+^b.^b sin2 t+2^a.^b sin t cos t|OP|=1+^a.^b sin 2t|OP|max=M=1+^a.^b at sin 2t=1t=π4At t=π4, OP=12(^a+^b)
Unit vector along OP at (t=π4)=^a+^b|^a+^b|


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