(D) The affix of the centroid $G$ is given by $g = \frac{z_1 + z_2 + z_3}{3}$.Since $z = 0$ is the midpoint of $AG$,the affix of the midpoint is $\fra

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(D) The affix of the centroid $G$ is given by $g = \frac{z_1 + z_2 + z_3}{3}$.Since $z = 0$ is the midpoint of $AG$,the affix of the midpoint is $\frac{g + z_1}{2} = 0$.Substituting the value of $g$,we get $\frac{\frac{z_1 + z_2 + z_3}{3} + z_1}{2} = 0$.Multiplying by $6$,we get $(z_1 + z_2 + z_3) + 3z_1 = 0$.Thus,$4z_1 + z_2 + z_3 = 0$.

Class 11 Mathematics 1 Complex Numbers and Quadratic Equations: Complex Numbers Geometry of complex numbers

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If ${z_1}, {z_2}, {z_3}$ are affixes of the vertices $A, B$ and $C$ respectively of a triangle $ABC$ having centroid at $G$ such that $z = 0$ is the midpoint of $AG$,then:

A
${z_1} + {z_2} + {z_3} = 0$
B
${z_1} + 4{z_2} + {z_3} = 0$
C
${z_1} + {z_2} + 4{z_3} = 0$
D
$4{z_1} + {z_2} + {z_3} = 0$

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1 Complex Numbers and Quadratic Equations: Complex Numbers

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Geometry of complex numbers

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If ${z_1}, {z_2}, {z_3}$ are affixes of the vertices $A, B$ and $C$ respectively of a triangle $ABC$ having centroid at $G$ such that $z = 0$ is the midpoint of $AG$,then:

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The points in the set $\{z \in \mathbb{C} : \arg \left(\frac{z-2}{z-6i}\right) = \frac{\pi}{2}\}$ (where $\mathbb{C}$ denotes the set of all complex numbers) lie on the curve which is a

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