0

Guest

Courses New

if Z1 and Z2are 2 complex nos show that they can form an equilatrel triangle with origin if and only if (Z1/Z2 + Z2/Z1)=1

if Z1 and Z2are 2 complex nos show that they can form an equilatrel triangle with origin if and only if (Z1/Z2 + Z2/Z1)=1

Grade:11

2 Answers

SAGAR SINGH - IIT DELHI
878 Points
13 years ago

Dear student,

LET A,B,C BE THE VERTICES REPRESENTED BY Z1,Z2,Z3...ABC IS EQUILATERAL HENCE


AB=BC=CA=R SAY............
AB=Z2-Z1...|Z2-Z1|=R
BC=Z3-Z2...|Z3-Z2|=R
CA=Z1-Z3...|Z1-Z3|=R

ANGLES..ABC=BCA=CAB=60 DEG.
ANGLE BETWEEN AB & BC IS ANGLE ABC..THAT IS BETWEEN Z2-Z1 AND Z3-Z2..ETC,..

HENCE IF WE TAKE
AB=Z2-Z1=RE^(iT)............I..........WHERE T IS SOME ANGLE....THEN
BC=Z3-Z2= RE^i(T+60)...........II
CA=Z1-Z3=RE^i(T-60).....................III
SQUARING EQNS.I,II,III AND ADDING
(Z2-Z1)^2+(Z3-Z2)^2+(Z1-Z3)^2=R^2[E^(2iT)+E^2i(T+60)+E^2i(T-60)]
2[Z1^2+Z2^2+Z3^2-Z1Z2-Z2Z3-Z3Z1]=(R^2)(E^2iT)[1+E^(120i)+E^(-120i)]

 

[1+COS(120)+iSIN(120)+COS(-120)+iSIN(-120)]=(R^2)(E^2iT)[1-0.5+iSIN(120)-0.5-iSIN(120)]=0
Z1^2+Z2^2+Z3^2-Z1Z2-Z2Z3-Z3Z1=0
Z1^2+Z2^2+Z3^2=Z1Z2+Z2Z3+Z3Z1


Ultimately it gives (Z1/Z2 + Z2/Z1)=1

 

Please feel free to ask your queries here. We are all IITians and here to help you in your IIT JEE preparation.

All the best.

Win exciting gifts by answering the questions on Discussion Forum. So help discuss any query on askiitians forum and become an Elite Expert League askiitian.

Now you score 5+15 POINTS by uploading your Pic and Downloading the Askiitians Toolbar  respectively : Click here to download the toolbar..

 

Askiitians Expert

Sagar Singh

B.Tech, IIT Delhi

sagarsingh24.iitd@gmail.com

mycroft holmes
272 Points
13 years ago

Think You Can Provide A Better Answer ?

Other Related Questions on Algebra

View all Questions »

ASK QUESTION

Get your questions answered by the expert for free

Answer and earn gift vouchers

Win Gift vouchers upto Rs 500/-

View courses by askIITians

Design classes One-on-One in your own way with Top IITians/Medical Professionals

Click Here Know More

Complete Self Study Package designed by Industry Leading Experts

Click Here Know More

Live 1-1 coding classes to unleash the Creator in your Child

Click Here Know More

a Complete All-in-One Study package Fully Loaded inside a Tablet!

Click Here Know More