A particle P starts from the point z0 = 1 + 2i, 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 z1. From z1 the particle moves √ 2 units in the direction of the vector hati + hatj and then it moves through an angle (π/2) in anti-clockwise direction on a circle with centre at origin, to reach a point z2. The point z2 is given by

Question

A particle P starts from the point z0 = 1 + 2i, 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 z1. From z1 the particle moves √ 2 units in the direction of the vector hati + hatj and then it moves through an angle (π/2) in anti-clockwise direction on a circle with centre at origin, to reach a point z2. The point z2 is given by (A) 6 + 7i (B) -7 + 6i (C) 7 + 6i (D) - 6 + 7i

Tardigrade Admin · Accepted Answer

<img class="img-fluid question-image" alt="image" src="https://cdn.tardigrade.in/img/question/chemistry/b44d1081f2790b011b9f2b85fe5506fd-.png" /> z2 prime=(6+√2 cos 45°, 5+√2 sin 45°)=(7,6)=7+6 i By rotation about (0,0) (z2/z2 prime)=ej π 2 ⇒ z2=z2 prime(ei (π/2)) =(7+6 i)( cos (π/2)+i sin (π/2)) =(7+6 i)(i)=-6+7 i

6 + 7i

-7 + 6i

7 + 6i

- 6 + 7i

z2​′=(6+2​cos45∘,5+2​sin45∘)=(7,6)=7+6i By rotation about (0,0) z2′​z2​​=ejπ2 ⇒z2​=z2​′(ei2π​) =(7+6i)(cos2π​+isin2π​) =(7+6i)(i)=−6+7i

$z_{2}^{'} = (6 + 2 cos 4 5^{\circ}, 5 + 2 sin 4 5^{\circ}) = (7, 6) = 7 + 6 i$
By rotation about $(0, 0)$
$\frac{z _{2}}{z _{2}^{'}} = e^{jπ 2}$
$\Rightarrow z_{2} = z_{2}^{'} (e^{i \frac{π}{2}})$
$= (7 + 6 i) (cos \frac{π}{2} + i sin \frac{π}{2})$
$= (7 + 6 i) (i) = - 6 + 7 i$