First slide

Geometry of complex numbers

Question

The vertices A,B,C of triangle are represented by the complex numbers $\frac{2}{\sqrt{3}} e^{i π / 2}, \frac{2}{\sqrt{3}} e^{- i π / 6}$ and $\frac{2}{\sqrt{3}} e^{- i 5 π / 6}$ respectively. Then which of following is NOT CORRECT?

Difficult

A

Area of $Δ A B C$ is $\frac{1}{\sqrt{3}}$ sq units
B

The equation of incircle of $Δ A B C$ is $| Z | = \frac{1}{\sqrt{3}}$
C

For any point ‘P’ on the incircle of $Δ A B C$ , then $P A^{2} + P B^{2} + P C^{2} = 5$
D

Equation of circle passing through A,B,C is $| Z - 1 | = \frac{2}{\sqrt{3}}$

Solution

The triangle is equilateral
Also $| Z_{1} | = | Z_{2} | = | Z_{3} | = \frac{2}{\sqrt{3}}$

$\Rightarrow$ Origin is circumcentre

$\Rightarrow$ Incentre is Origin

$\Rightarrow$ $r = \frac{1}{\sqrt{3}}$

$P A^{2} + P B^{2} + P C^{2} = {| Z - Z_{1} |}^{2} + {| Z - Z_{2} |}^{2} + {| Z - Z_{3} |}^{2}$

$= 3 (\frac{1}{3}) + 3 (\frac{4}{3}) - 0 = 5$

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