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arg(z1/z4) + arg(z2/z3) When z2, z4 Are Conjugates | JEE Advanced

JEE Maths question with a full step-by-step solution.

Question

z_2

and

z_4

are conjugates of

z_1

and

z_3

respectively, then

\arg\left(\frac{z_1}{z_4}\right) + \arg\left(\frac{z_2}{z_3}\right)

is equal to

\dfrac{\pi}{2}

\dfrac{3\pi}{2}

\pi

0

correct

Solution

Step 1: Combine the two arguments using

\arg A + \arg B = \arg(AB)

\arg\left(\frac{z_1}{z_4}\right) + \arg\left(\frac{z_2}{z_3}\right) = \arg\left(\frac{z_1 z_2}{z_3 z_4}\right).

Step 2: Put in the given conjugate relations

z_2 = \bar{z_1}

and

z_4 = \bar{z_3}

= \arg\left(\frac{z_1 \bar{z_1}}{z_3 \bar{z_3}}\right).

Step 3: Use

z\bar z = |z|^{2}

on the top and the bottom.

= \arg\left(\frac{\left|z_1\right|^{2}}{\left|z_3\right|^{2}}\right).

Step 4: This is a ratio of two positive real numbers, hence a positive real number, and a positive real number lies on the positive real axis. So its argument is

0 .

Answer: (4).

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