Let $z_{1}, z_{2}, z_{3}$ be the vertices of a triangle ABC. Then which of the following statements is correct?

see full answer

Your Exam Success, Personally Taken Care Of

1:1 expert mentors customize learning to your strength and weaknesses – so you score higher in school , IIT JEE and NEET entrance exams.

An Intiative by Sri Chaitanya

If $\frac{1}{z - z_{1}} + \frac{1}{z - z_{2}} + \frac{1}{z - z_{3}} = 0,$ where $z = \frac{|z_{1} + z_{2} + z_{3}|}{3},$ then ABC is an equilateral triangle.

If ABC is an equilateral triangle then $\frac{1}{z - z_{1}} + \frac{1}{z - z_{2}} + \frac{1}{z - z_{3}} = 0, w h e r e z = \frac{z_{1} + z_{2} + z_{3}}{3}$

If $| \begin{matrix} 1 & 1 & 1 \\ z_{1} & z_{2} & z_{3} \\ z_{2} & z_{3} & z_{1} \end{matrix} | = 0,$ then the triangle ABC is equilateral

If $| z_{1} | = | z_{2} | = | z_{3} | a n d z_{1} + z_{2} + z_{3} = 0,$ then the triangle ABC is equilateral

answer is B, A, C, D.

(Unlock A.I Detailed Solution for FREE)

Best Courses for You

JEE

NEET

Foundation JEE

Foundation NEET

CBSE

Detailed Solution

A necessary and sufficient condition for a triangle having vertices $z_{1}, z_{2} a n d z_{3}$ to form an equilateral triangle is $z_{1}^{2} + z_{2}^{2} + z_{3}^{2} = z_{1} z_{2} + z_{1} z_{3} + z_{2} z_{3}$
(A) and (B) will follow by performing some algebraic jugglery on the known condition given above to prove (D) note that $z_{1} + z_{2} + z_{3} = 0$ can be changed to
$\frac{1}{z_{1}} + \frac{1}{z_{2}} + \frac{1}{z_{3}} = 0 (∵ | z_{1} | = | z_{2} | = | z_{3} |)$