If complex numbers z1,z2 and z3 represents the vertices of an equilateral triangle such that |z1|=|z2|=|z3|, then

Question

If complex numbers z1,z2 and z3 represents the vertices of an equilateral triangle such that |z1|=|z2|=|z3|, then (A)  z1+z2+z3 le 0  (B)  z1+z2+z3 ge 0  (C)  z1+z2+z3=0  (D) None of these

Tardigrade Admin · Accepted Answer

Given, |z1|=|z2|=|z3| Three points lies on the circle, whose centre is origin. Since, these points are vertices of an equilateral triangle so orthocentre and circumcentre will be same. Hence, (z1+z2+z3/3)=0 ⇒ z1+z2+z3=0

Q. If complex numbers $z_{1}, z_{2}$ and $z_{3}$ represents the vertices of an equilateral triangle such that $∣ z_{1} ∣ = ∣ z_{2} ∣ = ∣ z_{3} ∣,$ then

$z_{1} + z_{2} + z_{3} \leq 0$

$z_{1} + z_{2} + z_{3} \geq 0$

$z_{1} + z_{2} + z_{3} = 0$

None of these

Q. If complex numbers z1​,z2​ and z3​ represents the vertices of an equilateral triangle such that ∣z1​∣=∣z2​∣=∣z3​∣, then

z1​+z2​+z3​≤0

z1​+z2​+z3​≥0

z1​+z2​+z3​=0

None of these

Given, ∣z1​∣=∣z2​∣=∣z3​∣ Three points lies on the circle, whose centre is origin. Since, these points are vertices of an equilateral triangle so orthocentre and circumcentre will be same. Hence, 3z1​+z2​+z3​​=0 ⇒ z1​+z2​+z3​=0

Q. If complex numbers $z_{1}, z_{2}$ and $z_{3}$ represents the vertices of an equilateral triangle such that $∣ z_{1} ∣ = ∣ z_{2} ∣ = ∣ z_{3} ∣,$ then

$z_{1} + z_{2} + z_{3} \leq 0$

$z_{1} + z_{2} + z_{3} \geq 0$

$z_{1} + z_{2} + z_{3} = 0$