Question

Let z₁ and z₂ be two non-zero complex numbers such that $\frac{z_{1}}{z_{2}}$+ $\frac{z_{2}}{z_{1}}$ = 1, then the origin and points represented by z₁ and z₂

• A. Lie on a straight line
• B. Form a right triangle
• C. Form an equilateral triangle
• D. None of these

Answer 1

Answer: (C)

Form an equilateral triangle

Answer 2

Sol. Let z = z₁/z₂, then z + 1/z = 1 Ž z² – z + 1 = 0

Ž z = $\frac{1 \pm \sqrt{3}i}{2}$ Ž $\frac{z_{1}}{z_{2}}$ = $\frac{1 \pm \sqrt{3}i}{2}$

If z₁ and z₂ are represented by A and B respectively and O be the origin, then

$\frac{OA}{OB}$= $\frac{|z_{1}|}{|z_{2}|}$ = $\left| \frac{1 \pm \sqrt{3}i}{2} \right|$ = $\sqrt{\frac{1}{4} + \frac{3}{4}}$= 1

Ž OA = OB

Also, $\frac{AB}{OB}$=$\frac{|z_{2} - z_{1}|}{|z_{2}|}$=$\left| 1 - \frac{z_{1}}{z_{2}} \right|$

=$\left| 1 - \left( \frac{1}{2} \pm \frac{\sqrt{3}}{2}i \right) \right|$ = $\left| \frac{1}{2} \mp \frac{\sqrt{3}}{2}i \right|$

=$\sqrt{\frac{1}{4} + \frac{3}{4}}$ = 1

Ž AB = OB

Thus, OA = OB = AB.

\ DOAB is an equilateral triangle.