Argument of Complex Number

If $\arg \left(z\right)<0,$ then $\arg \left(-z\right)-\arg \left(z\right)=$

Let $z$and $w$be two non-zero complex numbers such that  $\left|z\right|=\left|w\right|$  and  $\arg z+\arg w=\pi$  then $z$equals –

A complex number $z=1+i\sqrt{3}$.

The general argument of $z$ is

A complex number $z=\sqrt{3}+i$.

The general argument of $z$ is $2n\mathrm{\pi }+\frac{\mathrm{\pi }}{6}$, where $n$ is an integer.

A complex number $z=1-i$.

The argument of $\overline{z}$ is

A complex number $z=3+9i$.

Find the argument of $\overline{z}$.

If $z=x+iy$ and $\arg \left(\frac{z-1}{z+1}\right)=\frac{\pi }{4}$, find the locus of $\left(x, y\right)$.

Let $z_1=10+6 i, z_2=4+6 i$. If $z$ is any complex number such that, $\arg \left(\frac{z-z_1}{z-z_2}\right)=\frac{\pi }{4}$, then prove that $|z-7-9 i|=3\sqrt{2}$.

If $z_1=-\sqrt{3}+i$ and $z_2=i\sqrt{3}-1$, show that $arg\left(\frac{z_1}{z_2}\right)\ne arg\left(z_1\right)-arg\left(z_2\right)$

If $z_1=-\sqrt{3}+i$ and $z_2=i\sqrt{3}-1$, show that $arg\left(z_1{z}_2\right)\ne arg\left(z_1\right)+arg\left(z_2\right)$

If the argument of $\left(z-a\right)\left(\overline{z}-b\right)$ is equal to that of $\frac{\left(\sqrt{3}+i\right)\left(1+\sqrt{3}i\right)}{1+i}$, where $a, b$ are two real numbers and $\overline{z}$ is the complex conjugate of $z$, find the locus of $z$ in the Argand diagram. Find the values of $a$ and $b$ so the the locus becomes a circle having its centre at$\frac{1}{2}\left(3+i\right)$ .

If $z=x+i y$ and if $amp\left(\frac{z-1}{z+1}\right)=\frac{\pi }{4}$ then show that the locus (in complex plane) of $z$ is a circle.

Find the amplitude and modulus of $\frac{i}{1-i}$.

If $z=x+iy$ is a complex number in the $1$st quadrant of the Argand plane and if $\mathrm{amp}\left(\mathrm{z}-1\right)=\mathrm{amp}\left(\mathrm{z}+3 \mathrm{i}\right)$, then find the value of $xy-\left(x-1\right)\left(y+3\right)$.

Find the principle amplitude of $-3-3 i$

Find the principle amplitude of $-\sqrt{3 i}$

Find the principle amplitude of $\sqrt{3 i}$

$arg\left(\frac{1-i\sqrt{3}}{1+i\sqrt{3}}\right)=$

If $z$ is a purely imaginary number and $\mathrm{Im} \mathrm{z}<0$, then $\mathrm{amp} \mathrm{z}=$

The principle amplitude of $-1-i$ is