Andhra Pradesh Board Solutions for Exercise 3: Exercise 1(c)

If $z\ne 0,$ find $Argz+Arg\overline{z}$.

If $|z+a i|=|z-a i|$, then find the locus of $z$.

If $z=(x+iy)$ and if the point $P$ in the Argand plane represents $z$, then describe the locus of $P$ satisfying the equation $|2z-3|=7$.

If $z=(x+iy)$ and if the point $P$ in the Argand plane represents $z$, then describe geometrically the locus of $P$ satisfying the equation $|z{|}^2=4Re\left(z+2\right)$.

If $z=(x+iy)$ and if the point $P$ in the Argand plane represents $z$, then describe geometrically the locus of $P$ satisfying the equation $|z+i{|}^2-|z-i{|}^2=2$.

If $z=(x+iy)$ and if the point $P$ in the Argand plane represents $z$, then describe geometrically the locus of $P$ satisfying the equation $|z+4i|+|z-4i|=10$.

If $z_1, z_2$ are two nonzero complex numbers satisfying $\left|z_1+z_2\right|=\left|z_1\right|+\left|z_2\right|$, then show that $Arg{z}_1-Arg{z}_2=0$.

If $z=x+iy$ and the point $P$ represents $z$ in the Argand plane and $\left|\frac{z-a}{z+a}\right|=1,Re\left(a\right)\ne 0,$ then find the locus of $P$.