Question

If ${z_1},{z_2}$ and ${z_3}$ are complex numbers such that $\left| {{z_1}} \right| = \left| {{z_2}} \right| = \left| {{z_3}} \right| = \left| {\dfrac{1}{{{z_1}}} + \dfrac{1}{{{z_2}}} + \dfrac{1}{{{z_3}}}} \right| = 1$ then $\left| {{z_1} + {z_2} + {z_3}} \right|$is :
A. Equal to $1$
B. Less than $1$
C. Greater than $3$
D. Equal to $3$

Answer 1

For solving this particular problem, we first take the given expression that is $|{z_1}| = |{z_2}| = |{z_3}| = 1$ , then we will square this expression . then we will use the relation we have as ${\left| {{z_1}} \right|^2} = {z_1}\overline {{z_1}}$ , multiplication of the complex number with the conjugate of the complex number we get magnitude which represents the distance of the complex number from the origin. Then we will get, ${z_1}\overline {{z_1}} = {z_2}\overline {{z_2}} = {z_3}\overline {{z_3}} = 1$ , then substitute the result in the equation $\left| {\dfrac{1}{{{z_1}}} + \dfrac{1}{{{z_2}}} + \dfrac{1}{{{z_3}}}} \right| = 1$ , and try to manipulate this equation to get the value of $\left| {{z_1} + {z_2} + {z_3}} \right|$ .

Complete solution step by step:
Now we know that,
$|{z_1}| = |{z_2}| = |{z_3}| = 1$ (given)
Now Squaring the given expression, we get the following ,
$|{z_1}{|^2} = |{z_2}{|^2} = |{z_3}{|^2} = {1^2}......(1)$
Now we know that ${\left| {{z_1}} \right|^2} = {z_1}\overline {{z_1}}$ , multiplication of the
complex number with the conjugate of the complex number we get magnitude which represents the distance of the complex number from the origin. we get, ${z_1}\ overline {{z_1}} = {z_2}\overline {{z_2}} = {z_3}\overline {{z_3}} = 1$ ,
Now,
$\Rightarrow 1 = \left| {\dfrac{1}{{{z_1}}} + \dfrac{1}{{{z_2}}} + \dfrac{1}{{{z_3}}}} \right| \\\ = \left| {\dfrac{{{z_1}\overline {{z_1}} }}{{{z_1}}} + \dfrac{{{z_2}\overline {{z_2}} }}{{{z_2}}} + \dfrac{{{z_3}\overline {{z_3}} }}{{{z_3}}}} \right| \\\ = \left| {\overline {{z_1}} + \overline {{z_2}} + \overline {{z_3}} } \right| \\\ = \overline {\left| {{z_1} + {z_2} + {z_3}} \right|} \\\ = \left| {{z_1} + {z_2} + {z_3}} \right| \\\ = 1 \\\$
Hence we can say that the value of $\left| {{z_1} + {z_2} + {z_3}} \right|$ is equal to one.
Therefore, option A is correct.

Formula Used:
For solving this particular solution ,we used the following relationship ,
${\left| {{z_1}} \right|^2} = {z_1}\overline {{z_1}}$ , here $\overline {{z_1}}$ means the
conjugate of ${z_1}$ .

Note: As we know that $z = x + yi$ , which is the representation of the complex number. And $z = x - yi$ , is the conjugate of the complex number. Now multiplication of the complex number with the conjugate of the complex number we get magnitude which represents the distance of the complex number from the origin.