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Introduction

Let z be a co...

Question

Let $z$ be a complex number such that the principal value of argument, $arg z > 0$ . Then $arg z - arg (- z)$ is

\frac{π}{2}

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\pm π

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π

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- π

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Solution

The correct option is C $π$
Principal Value
Quad $I = θ$
Quad $I I = π - θ$
Quad $I I I = θ - π$
Quad $I V = - θ$

If $arg z = θ > 0$
$\Rightarrow$ Complex number $(z)$ lies in the $I$ or $I I$ quadrant.
$\Rightarrow - z$ lies in the $I I I$ or $I V$ quadrant.

Case:1-
$z$ lies in $I$ quadrant $\Rightarrow arg z = θ$
$\Rightarrow - z$ lies in $I I I$ quadrant $\Rightarrow arg (- z) = θ - π$
$∴ arg z - arg (- z) = π$

Case:2-
$z$ lies in $I I$ quadrant $\Rightarrow arg z = π - θ$
$\Rightarrow - z$ lies in $I V$ quadrant $\Rightarrow arg (- z) = - θ$
$∴ arg z - arg (- z) = π$

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