A community for students.
Here's the question you clicked on:
 0 viewing
mathslover
 2 years ago
If \(z_1\) and \(z_2\) are two complex numbers such that \(\frac{z_1  z_2}{z_1 + z_2} =1 \) , prove that, \(\frac{iz_1}{z_2}=k\) is a real number. Find the angle between the lines from the origin to the points \(z_1 + z_2 \) and \(z_1  z_2\) in terms of k.
mathslover
 2 years ago
If \(z_1\) and \(z_2\) are two complex numbers such that \(\frac{z_1  z_2}{z_1 + z_2} =1 \) , prove that, \(\frac{iz_1}{z_2}=k\) is a real number. Find the angle between the lines from the origin to the points \(z_1 + z_2 \) and \(z_1  z_2\) in terms of k.

This Question is Closed

mathslover
 2 years ago
Best ResponseYou've already chosen the best response.2@AravindG @amistre64 @ghazi @hartnn

AravindG
 2 years ago
Best ResponseYou've already chosen the best response.1oh too small latex ..my eye hurts

amistre64
 2 years ago
Best ResponseYou've already chosen the best response.0isnt a general complex number: a + bi ? or do we need to define it in trig terms?

mathslover
 2 years ago
Best ResponseYou've already chosen the best response.2If \(\large{z_1}\) and \(\large{z_2}\) are two complex numbers such that \(\large{\frac{z_1  z_2}{z_1 + z_2} = 1 }\) , prove that , \(\large{\frac{i z_1}{z_2} = k }\) , where k is a real number. Find the angle between the lines from the origin to the points \(\large{z_1 + z_2}\) and \(\large{z_1  z_2}\) in terms of k... It did hurt me too :)

mathslover
 2 years ago
Best ResponseYou've already chosen the best response.2Both conditions can be applied... @amistre64

mathslover
 2 years ago
Best ResponseYou've already chosen the best response.2well I started like this : \[\large{ \frac{\frac{z_1}{z_2} 1}{\frac{z_1}{z_2} +1} = 1}\] \[\large{\textbf{or} \space \frac{z_1}{z_2} 1 = \frac{z_1}{z_2}+1 }\]

mathslover
 2 years ago
Best ResponseYou've already chosen the best response.2I am thinking of squaring both sides, should I go for it ?

amistre64
 2 years ago
Best ResponseYou've already chosen the best response.0\[\frac{(a+bi)(n+mi)}{(a+bi)+(n+mi)}=1\] \[(a+bi)(n+mi)=(a+bi)+(n+mi)\] \[(n+mi)=(n+mi)\] \[0=2(n+mi)\] just a thought ....

AravindG
 2 years ago
Best ResponseYou've already chosen the best response.1i think I made a mistake there ..wait lemme think

mathslover
 2 years ago
Best ResponseYou've already chosen the best response.2But amistre where did modulus go ? In LHS ...

mathslover
 2 years ago
Best ResponseYou've already chosen the best response.2\[\large{\frac{z_1z_2}{z_1+z_2} = 1 \ne \frac{z_1 z_2}{z_1+z_2} =1 }\]

mathslover
 2 years ago
Best ResponseYou've already chosen the best response.2^ not necessary that : (a+bi)(n+mi) / (a+bi)+(n+mi) = 1 ...

shubhamsrg
 2 years ago
Best ResponseYou've already chosen the best response.0You can do it easily by interpretting is geometrically.

amistre64
 2 years ago
Best ResponseYou've already chosen the best response.0im just not that attuned to complex calculations :)

AravindG
 2 years ago
Best ResponseYou've already chosen the best response.1z1+z2 =z1z2 this means lenth of the vectors z1+z2 and z1z2 are equal Now think of z1 and z2 as two vectors z1+z2 is one diagonal of the parallelogram ,z1z2 is the other diagonal Given the diagonal lengths are equal thus the parallelogram is a rectangle So angle between z1 and z2 is \(\dfrac{\pi}{2}\)

AravindG
 2 years ago
Best ResponseYou've already chosen the best response.1;) geometry is wonderful

mathslover
 2 years ago
Best ResponseYou've already chosen the best response.2can you draw that parallelogram please @AravindG

mathslover
 2 years ago
Best ResponseYou've already chosen the best response.2I am confused that whether that parallelogram is possible or not with two diagonals as : z1 + z2 and z1z2

hartnn
 2 years ago
Best ResponseYou've already chosen the best response.1\(\huge \frac{(a+bi)(n+mi)}{(a+bi)+(n+mi)}=1 \\ (an)^2+(bm)^2=(a+n)^2+(b+n)^2 \\ \implies an+bm=0\) \(\huge \frac{i(a+bi)}{(n+mi)}=\frac{i(a+bi)}{(n+mi)}\dfrac{(nmi)}{(nmi)}= \\ \huge =\dfrac{i(an+bminb+ami)}{(n^2+m^2)}=\dfrac{i^2(0+am+nb)}{n^2+m^2}= \\ =k\)

mathslover
 2 years ago
Best ResponseYou've already chosen the best response.2well right :) I got it ..

mathslover
 2 years ago
Best ResponseYou've already chosen the best response.2But what about the second one..

mathslover
 2 years ago
Best ResponseYou've already chosen the best response.2I did the first one like this : \[\large{\frac{z_1}{z_2}  1 =  \frac{z_1}{z_2} + 1}\] Squaring both sides : \[\large{\frac{z_1}{z_2}^2 + 1  2Re (\frac{z_1}{z_2}) = \frac{z_1}{z_2}^2 + 1 + 2 Re(\frac{z_1}{z_2})}\] \[\large{4Re(\frac{z_1}{z_2}) = 0}\] \[\large{Re(\frac{z_1}{z_2}) = 0}\] therefore (z_1)/z_2 is purely imaginary number. Therefore \[\large{\frac{z_1}{z_2} = i \frac{z_1}{z_2} = k}\] where k is a real number

AravindG
 2 years ago
Best ResponseYou've already chosen the best response.1@mathslover where do you have confusion in my working ?

mathslover
 2 years ago
Best ResponseYou've already chosen the best response.2I am confused whether that parallelogram is possible or not?

mathslover
 2 years ago
Best ResponseYou've already chosen the best response.2can you draw that for me please?

AravindG
 2 years ago
Best ResponseYou've already chosen the best response.1just consider z1 and z2 as two vectors .... P.S. I am bad at drawing

mathslover
 2 years ago
Best ResponseYou've already chosen the best response.2dw:1361019499183:dw is it right diagram ?

mathslover
 2 years ago
Best ResponseYou've already chosen the best response.2instead of drawing can you tell me that what are co ordinates of the parallelogram : like A(z_1) B(z_2) C(z_(1)) D(z_(2)) ...

AravindG
 2 years ago
Best ResponseYou've already chosen the best response.1i would say this : dw:1361019666403:dw

AravindG
 2 years ago
Best ResponseYou've already chosen the best response.1gosh u made me draw that ...:P

mathslover
 2 years ago
Best ResponseYou've already chosen the best response.2dw:1361019850236:dw
Ask your own question
Sign UpFind more explanations on OpenStudy
Your question is ready. Sign up for free to start getting answers.
spraguer
(Moderator)
5
→ View Detailed Profile
is replying to Can someone tell me what button the professor is hitting...
23
 Teamwork 19 Teammate
 Problem Solving 19 Hero
 Engagement 19 Mad Hatter
 You have blocked this person.
 ✔ You're a fan Checking fan status...
Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.