A particle $P$ starts from the point $z_0 = 1 + 2i$, where $i = \sqrt{-1}$. It moves 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

A particle P starts from the point z₀ = 1 + 2i, where i = √-1. It moves 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 î + ĵ and then it moves through an angle π/2 in anticlockwise direction on a circle with centre at origin, to reach a point z₂. The point z₂ is given by

$\text{Given: } z_0 = 1 + 2i$

$z_0 \to z_1:\ +5\ \text{(real)},\ +3\ \text{(imaginary)}$

$z_1 \to z_2:\ \sqrt{2}\ \text{along}\ \hat{i} + \hat{j},\ \text{then rotate } \dfrac{\pi}{2}\ \text{anticlockwise}$

$\text{To find: } z_2$

$\text{Argand plane}$

$z_0 = 1 + 2i$

$\text{horizontal: } x_0 + 5 = 1 + 5 = 6$

$\text{vertical: } y_0 + 3 = 2 + 3 = 5$

$z_1 = 6 + 5i$

$\hat{u} = \dfrac{\hat{i} + \hat{j}}{\sqrt{2}}$

$\text{displacement} = \sqrt{2}\,\hat{u} = \hat{i} + \hat{j} = 1 + i$

$Q = z_1 + (1+i) = 6 + 5i + 1 + i = 7 + 6i$

$\text{rotate } \dfrac{\pi}{2} \text{ anticlockwise} \iff \text{multiply by } e^{i\pi/2} = i$

$z_2 = i(7 + 6i) = 7i + 6i^2 = -6 + 7i$

$|z_2| = \sqrt{(-6)^2 + 7^2} = \sqrt{36 + 49} = \sqrt{85} = |7 + 6i|$

$z_2 = -6 + 7i \quad \Rightarrow\ \text{option (4)}$