First slide

Geometry of complex numbers

Question

Statement 1 : The complex equation |Z+ 2 - i| = |Z + i – 2| represents a straight line.
Statement II : If Z₀ is the circumcentre of an equilateral triangle with vertices Z₁, Z₂, Z₃ then $Z_{1}^{2} + Z_{2}^{2} + Z_{3}^{2} = 2. Z_{0}^{2}$ .

Easy

A

Only I is true
B

Only II Is true
C

Both I, II are true
D

Neither I, nor II are true

Solution

$| x + 2 + i (y - 1) | = | (x - 2) + i (y + 1) |$ $\Rightarrow {(x + 2)}^{2} + {(y - 1)}^{2} = {(x - 2)}^{2} + {(y - 1)}^{2}$
Which is a straight line. By synopsis, $Z_{1}^{2} + Z_{2}^{2} + Z_{3}^{3} = 3 Z_{0}^{2}$ .

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