The area of the triangle vertices $\mathrm{A}(\mathrm{z}), \mathrm{B}(\mathrm{iz})$ and $\mathrm{C}(\mathrm{z}+\mathrm{iz})$ is :

JEE Main · 2021 · 17 Mar 2021 · Shift 1 · Complex Numbers

The area of the triangle vertices

A (z), B (iz)

and

C (z + iz)

is :

Answer: (A)

Let

z = x + i y

. Then

A (z) = (x, y)

B (i z) = (- y, x)

, and

C (z + i z) = (x - y, y + x)

. The area of a triangle with vertices

(x_{1}, y_{1}), (x_{2}, y_{2}), (x_{3}, y_{3})

\frac{1}{2} ∣ x_{1} (y_{2} - y_{3}) + x_{2} (y_{3} - y_{1}) + x_{3} (y_{1} - y_{2}) ∣

. Substituting: Area

= \frac{1}{2} ∣ x (x - (y + x)) + (- y) (y + x - y) + (x - y) (y - x) ∣ = \frac{1}{2} ∣ x (- y) + (- y) (x) + (x - y) (y - x) ∣ = \frac{1}{2} ∣ - x y - x y - (x - y)^{2} ∣ = \frac{1}{2} ∣ - 2 x y - x^{2} + 2 x y - y^{2} ∣ = \frac{1}{2} ∣ - (x^{2} + y^{2}) ∣ = \frac{1}{2} ∣ z ∣^{2}

JEE Main · 2021 · 17 Mar 2021 · Shift 1 · Single correct option