If z1, z2 and z3 are the vertices of a triangle in the argand plane such that |z1 - z2|=|z1 - z3|, then |arg ((2 z1 - z2 - z3/z3 - z2))| is

Question

If z1, z2 and z3 are the vertices of a triangle in the argand plane such that |z1 - z2|=|z1 - z3|, then |arg ((2 z1 - z2 - z3/z3 - z2))| is (A) (π /3) (B) 0 (C) (π /2) (D) (π /6)

Tardigrade Admin · Accepted Answer

<img class="img-fluid question-image" alt="Solution" src="https://cdn.tardigrade.in/q/nta/m-jkxf4ojrzrc6.png" /> (z1 - z2/z3 - z2)=(|z1 - z2|/|z3 - z2|)ei θ (z1 - z2/z2 - z3)=(|z1 - z3|/|z2 - z3|)e- i θ ⇒ ((z1 - z2/z3 - z2) + (z1 - z3/z3 - z2))= purely imaginary number arg((2 z1 - z2 - z3/z3 - z2))=±(π /2)

Q. If $z_{1}, z_{2}$ and $z_{3}$ are the vertices of a triangle in the argand plane such that $∣ z_{1} - z_{2} ∣ = ∣ z_{1} - z_{3} ∣,$ then $∣ ∣ a r g (\frac{2 z _{1} - z _{2} - z _{3}}{z _{3} - z _{2}}) ∣ ∣$ is

$\frac{π}{3}$

$0$

$\frac{π}{2}$

$\frac{π}{6}$

Q. If z1​,z2​ and z3​ are the vertices of a triangle in the argand plane such that ∣z1​−z2​∣=∣z1​−z3​∣, then ∣∣​arg(z3​−z2​2z1​−z2​−z3​​)∣∣​ is

3π​

0

2π​

6π​

z3​−z2​z1​−z2​​=∣z3​−z2​∣∣z1​−z2​∣​eiθz2​−z3​z1​−z2​​=∣z2​−z3​∣∣z1​−z3​∣​e−iθ ⇒(z3​−z2​z1​−z2​​+z3​−z2​z1​−z3​​)= purely imaginary number arg(z3​−z2​2z1​−z2​−z3​​)=±2π​

Q. If $z_{1}, z_{2}$ and $z_{3}$ are the vertices of a triangle in the argand plane such that $∣ z_{1} - z_{2} ∣ = ∣ z_{1} - z_{3} ∣,$ then $∣ ∣ a r g (\frac{2 z _{1} - z _{2} - z _{3}}{z _{3} - z _{2}}) ∣ ∣$ is

$\frac{π}{3}$

$0$

$\frac{π}{2}$

$\frac{π}{6}$