IIT JEE , AIEEE Forum - Mathematics , Algebra - if z1 & z2 r 2 complex nums.satisfying eq. |z1+z2/z1-z2|=1 then z1/z2 is 0/imaginry how?

bhargava

if z1 & z2 r 2 complex nums.satisfying eq. |z1+z2/z1-z2|=1 then z1/z2 is 0/imaginry how?

pramod6990

do one thing bring the mod to the other side and square both sides up...to get

(/Z1+Z2/)² = (/Z1-Z2/)²and then u can use the fact that /Z/ ² = Z {(Z)bar}.....to get the reqd. result...

rate if useful...

krish1092

If it is useful then please rate me

edison

I think the approcah suggested by pramod should work.

hsbhatt

|z_1+z_2| =|z_1 - z_2|

From the analogy with vectors, this means that z_1, z_2 are perpendicular to each other.

Alternatively,

$\left| \frac{z_1+z_2}{z_1-z_2} \right | = 1\\ \\<br/>\Rightarrow \frac{z_1+z_2}{z_1-z_2} = \cos \theta + i \sin \theta \\ \\<br/>$

Using componendo-dividendo,

$\frac{z_1}{z_2} = \frac{1+\cos \theta + i \sin \theta}{\cos \theta -1 + i \sin \theta} = -i \cot \frac{\theta}{2}$