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JEE Advanced2008MathematicsVector AlgebraActual

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 O P (where, O is the origin) is given by a t+ b t . When P is farthest from origin O , let M be the length of O P and u be the unit vector along O P . Then,

Options

  1. Au = a + b | a + b | and M=(1+ a b )^ 1 / 2
  2. Bu = a - b | a - b | and M=(1+ a b )^ 1 / 2
  3. Cu = a + b | a + b | and M=(1+2 a b )^ 1 / 2
  4. Du = a - b | a - b | and M=(1+2 a b )^ 1 / 2

Correct answer

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

Step-by-step solution

O P = a t+ b t aligned & | O P |= ( a a ^2 t+ b b ^2 t+2 a b t t ) & | O P |= 1+2 a b t t & | O P |= 1+ a b 2 t & | O P |_ = 1+ a b at 2 t=1 t= 4 & O P ( at t= 4 )= 1 2 ( a + b ) & aligned Unit vector along O P at (t= 4 )= a + b | a + b |

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