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Question: The complex numbers \(z_{1},z_{2}\) and \(z_{3}\) satisfying \(\frac{z_{1} - z_{3}}{z_{2} - z_{3}} ...

The complex numbers z1,z2z_{1},z_{2} and z3z_{3} satisfying

z1−z3z2−z3=1−i32\frac{z_{1} - z_{3}}{z_{2} - z_{3}} = \frac{1 - i\sqrt{3}}{2}are the vertices of a triangle which is

A

Of area zero

B

Right-angled isosceles

C

Equilateral

D

Obtuse-angled isosceles

Answer

Equilateral

Explanation

Solution

Sol. Taking mod of both sides of given relation

∣z1−z3z2−z3∣=∣12−i32∣=14+34=1\left| \frac{z_{1} - z_{3}}{z_{2} - z_{3}} \right| = \left| \frac{1}{2} - i\frac{\sqrt{3}}{2} \right| = \sqrt{\frac{1}{4} + \frac{3}{4}} = 1So, ∣z1−z3∣=∣z2−z3∣|z_{1} - z_{3}| = |z_{2} - z_{3}|. Also, amp (z1−z3z2−z3)\left( \frac{z_{1} - z_{3}}{z_{2} - z_{3}} \right) = tan⁡−1(−3)=−π3\tan^{- 1}{}( - \sqrt{3}) = - \frac{\pi}{3} or

amp(z2−z3z1−z3)=π3amp\left( \frac{z_{2} - z_{3}}{z_{1} - z_{3}} \right) = \frac{\pi}{3} or ∠z2z3z1=60∘\angle z_{2}z_{3}z_{1} = 60{^\circ}

∴ The triangle has two sides equal and the angle between the equal sides =60∘= 60{^\circ}. So it is equilateral.