If {z_1},{z_2},{z_3} are complex numbers such | vedclass

Name: PDF Generator App | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(a)$1 = \left| {\frac{1}{{{z_1}}} + \frac{1}{{{z_2}}} + \frac{1}{{{z_3}}}} \right|$$ = \left| {\frac{{{z_1}{{\bar z}_1}}}{{{z_1}}} + \frac{{{z_2}{{\ba

Vedclass · Accepted Answer

Equal to $1$

Vedclass · Answer

(a)$1 = \left| {\frac{1}{{{z_1}}} + \frac{1}{{{z_2}}} + \frac{1}{{{z_3}}}} \right|$$ = \left| {\frac{{{z_1}{{\bar z}_1}}}{{{z_1}}} + \frac{{{z_2}{{\bar z}_2}}}{{{z_2}}} + \frac{{{z_3}{{\bar z}_3}}}{{{z_3}}}} \right|$ 

$(\,\,\,|{z_1}{|^2} = 1 = {z_1}{\overline z _1},{\rm{etc}})$
$ = \,|{\bar z_1} + {\bar z_2} + {\bar z_3}|\, = \,|\overline {{z_1} + {z_2} + {z_3}} |\, = \,|{z_1} + {z_2} + {z_3}|$

$(\because \,\,\,|{\bar z_1}| = |{z_1}|)$

If ${z_1},{z_2},{z_3}$ are complex numbers such that $|{z_1}|\, = \,|{z_2}|\, = $ $\,|{z_3}|\, = $ $\left| {\frac{1}{{{z_1}}} + \frac{1}{{{z_2}}} + \frac{1}{{{z_3}}}} \right| = 1\,,$ then${\rm{ }}|{z_1} + {z_2} + {z_3}|$ is

Similar Questions

If ${z_1},{z_2} \in C$, then $amp\,\left( {\frac{{{{\rm{z}}_{\rm{1}}}}}{{{{{\rm{\bar z}}}_{\rm{2}}}}}} \right) = $

If ${z_1} = a + ib$ and ${z_2} = c + id$ are complex numbers such that $|{z_1}| = |{z_2}| = 1$ and $R({z_1}\overline {{z_2}} ) = 0,$ then the pair of complex numbers ${w_1} = a + ic$ and ${w_2} = b + id$ satisfies

Find the modulus and the argument of the complex number $z=-1-i \sqrt{3}$.

If complex numbers $(x -2y) + i(3x -y)$ and $(2x -y) + i(x -y + 6)$ are conjugates of each other, then $|x + iy|$ is $(x,y \in R)$

The sum of amplitude of $z$ and another complex number is $\pi $. The other complex number can be written