Question

If ${Z_1},{Z_2}$ and ${Z_3},{Z_4}$ are two pair of complex conjugate numbers, then $\arg \left( {\dfrac{{{Z_1}}}{{{Z_4}}}} \right) + \arg \left( {\dfrac{{{Z_2}}}{{{Z_3}}}} \right)$ equals:
A. $0$
B. $\pi $
C. $\dfrac{\pi }{2}$
D. $\dfrac{{3\pi }}{2}$

Answer 1

Hint: According to given in the question we have to determine the value of $\arg \left( {\dfrac{{{Z_1}}}{{{Z_4}}}} \right) + \arg \left( {\dfrac{{{Z_2}}}{{{Z_3}}}} \right)$if${Z_1},{Z_2}$and ${Z_3},{Z_4}$are two pair of complex conjugate numbers So, first of all as we know that ${Z_1},{Z_2}$and ${Z_3},{Z_4}$are conjugate complete numbers so, we can determine the value of $\arg \left( {\dfrac{{{Z_1}}}{{{Z_4}}}} \right) + \arg \left( {\dfrac{{{Z_2}}}{{{Z_3}}}} \right)$.
Now, we have to use the formula as mentioned below for complex functions,

Formula used: $ \Rightarrow \arg (a) + \arg (b) = \arg (a.b)................(A)$
So, with the help of the formula (A) above, and we have to determine the conjugate of ${Z_1}$ and ${Z_3}$ then we can determine the required solution.

Complete step-by-step solution:
Step 1: First of all as we know that ${Z_1},{Z_2}$and ${Z_3},{Z_4}$are conjugate complete numbers so, we can determine the value of $\arg \left( {\dfrac{{{Z_1}}}{{{Z_4}}}} \right) + \arg \left( {\dfrac{{{Z_2}}}{{{Z_3}}}} \right)$ as mentioned in the solution hint.
$ \Rightarrow {Z_2} = \overline {{Z_1}} $and,
$ \Rightarrow {Z_4} = \overline {{Z_3}} $
Step 2: Now, we have to use the formula (A) to solve the given complex expression as,
$ \Rightarrow \arg \left( {\dfrac{{{Z_1}}}{{{Z_4}}}} \right) + \arg \left( {\dfrac{{{Z_2}}}{{{Z_3}}}} \right) = \arg \left( {\dfrac{{{Z_1}}}{{{Z_4}}}} \right)\left( {\dfrac{{{Z_2}}}{{{Z_3}}}} \right)$……………….(1)
Step 3: Now, we have to determine the conjugate of ${Z_1},{Z_3}$in the expression (1) as obtained in the solution step 2.
$ = \arg \left( {\dfrac{{{Z_1}}}{{{Z_3}}}} \right)\left( {\dfrac{{\overline {{Z_1}} }}{{{Z_3}}}} \right)$
\[
   = \arg \left( {\dfrac{{{Z_1}\overline {{Z_1}} }}{{{Z_3}\overline {{Z_3}} }}} \right) \\
   = \arg \left( {\dfrac{{{{\left| {{Z_1}} \right|}^2}}}{{{{\left| {{Z_3}} \right|}^2}}}} \right) \\
   = 0
 \]
Which is purely real.
Hence, with the help of the formula (A) above, we have determined the value of the given complex expression which is $\arg \left( {\dfrac{{{Z_1}}}{{{Z_4}}}} \right) + \arg \left( {\dfrac{{{Z_2}}}{{{Z_3}}}} \right)$= 0.

Therefore, option (A) is correct.

Note: If the given complex number is Z then we can represent it as $(a + ib)$ and its conjugate is $\overline Z $ which is as $(a - ib)$ and where, a is a real number and I is imaginary.
If two complex number and such that $\arg A + \arg B$ then we can represent it as the multiplication of the complex numbers as $\arg (A.B)$