Question

Assertion (A): Area of triangle formed by $1+i,-1+i,2i$ is 1 square unit.

Reason (R): Area of triangle formed by the complex numbers $z,iz,z+iz$ is
$\frac{1}{2}|z{|}^2$

• A. Both A and R are true and R is the correct explanation of A
• B. Both A and R are true and R is the correct explanation of A
• C. A is false, R is true
• D. A is true R is false

Answer 1

The correct option is A Both A and R are true and R is the correct explanation of A

Reason
Clearly $A\left(z\right),B\left(iz\right),C(z+iz)andD\left(0\right)$
are all vertices of a parallelogram
as, $\left|AD\right|=\left|BD\right|=\left|CB\right|=\left|AC\right|$
also,
AD and BD sides subtend
$\frac{\pi }{2}=90atD(B=iz=e^i\frac{\pi }{2}z)rotatezby90$
So, ADBC is a square
$area=(side{)}^2={\left|z\right|}^2$
$area\left(\Delta ACB\right)=\frac{1}{2}|z{|}^2$
Assertion,
put $z=1+i$
$iz=-1+i$
$z+iz=2i$
$area=\frac{1}{2}|z{|}^2$
$=\frac{1}{2}\times 2$
$=1unit$