A particle $P$ starts from the point $z_{0}=1+2 i$, where $i=\sqrt{-1}$. It moves first horizontally away from origin by 5 units and then vertically away from origin by 3 units to reach a point $z_{1}$. From $z_{1}$ the particle moves $\sqrt{2}$ units in the direction of the vector $\hat{i}+\hat{j}$ and then it moves through an angle $\frac{\pi}{2}$ in anticlockwise direction on a circle with centre at origin, to reach a point $z_{2}$. The point $z_{2}$ is given by

JEE Advanced · 2008 · 13 Apr 2008 · Paper 2 · Complex Numbers

A particle

P

starts from the point

z_{0} = 1 + 2 i

, where

i = - 1

. It moves first horizontally away from origin by 5 units and then vertically away from origin by 3 units to reach a point

z_{1}

. From

z_{1}

the particle moves

2

units in the direction of the vector

\hat{i} + \hat{j}

and then it moves through an angle

\frac{π}{2}

in anticlockwise direction on a circle with centre at origin, to reach a point

z_{2}

. The point

z_{2}

is given by

Answer: (D)

Starting from

z_{0} = 1 + 2 i

, moving horizontally away from origin by 5 units means moving in the positive real direction:

z_{1} = (1 + 5) + (2 + 3) i = 6 + 5 i

(after also moving vertically by 3 units). From

z_{1} = 6 + 5 i

, moving

2

units in the direction of

\hat{i} + \hat{j}

(which is at 45° to the real axis) adds

1 + i

to get

7 + 6 i

. This point has modulus

49 + 36 = 85

. Rotating by

\frac{π}{2}

anticlockwise multiplies by

i

(7 + 6 i) \cdot i = 7 i + 6 i^{2} = - 6 + 7 i

. Hence

z_{2} = - 6 + 7 i

JEE Advanced · 2008 · 13 Apr 2008 · Paper 2 · Single correct option