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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
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
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
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.
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Now you score 5+15 POINTS by uploading your Pic and Downloading the Askiitians Toolbar respectively : Click here to download the toolbar..
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Sagar Singh
B.Tech, IIT Delhi
sagarsingh24.iitd@gmail.com
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