Given :   A(2 + i) is rotated about origin with angle pi/4   in anticlockwise direction to reach at B

To Find : B

Solution:

A  = 2  +  i

| A |² = 2²  + 1²  = 5

Angle = Tan⁻¹( 1/2)

New Angle = Tan⁻¹( 1/2)  + π/4

Tan (Tan⁻¹( 1/2)  + π/4)

using Tan(A + B) = (Tan A + Tan B ) /(1 - TanATanB)

Tan (Tan⁻¹( 1/2)) = 1/2  and Tan π/4 = 1

= (1/2 + 1)/(1 - (1/2)1)

= (3/2)/(1/2)

= 3

New Position = x  + iy

y/x  = 3

=> y = 3x

x² + y² = 5

=> x² + (3x)² = 5

=> x² + 9x² = 5

=> 10x² = 5

=> x² = 1/2

=> x = 1/√2

y = 3/√2

1/√2 + i 3/√2

= (1 + 3i)/√2

B is (1 + 3i)/√2

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