Question

$P$ represents the variable complex number $z$. Find the locus of $P$, if Re$\left(\frac{z+1}{z+i}\right)=1$

Answer 1

Let $z=x+iy$

then $\frac{z+1}{z+i}=\frac{x+iy+1}{x+iy+i}=\frac{(x+1)+iy}{x+i(y+1)}$

$=\frac{(x+1)+iy}{x+i(y+1)}\times \frac{x-i(y+1)}{x+i(y+1)}$

$=\frac{x(x+1)+y(y+1)+i(yx-xy-x-y-1)}{x^2+(y+1{)}^2}$

$=\frac{x(x+1)+y(y+1)+i(-x-y-1)}{x^2+(y+1{)}^2}$

Given that $Re\left(\frac{z+1}{z+i}\right)=1$

$\therefore \frac{x(x+1)+y(y+1)}{x^2+(y+1{)}^2}=1$

$\Rightarrow x^2+y^2+x+y=x^2+y^2+2y+1$

$\Rightarrow x-y=1$

The locus of $P$ is $x-y=1$.