Question

Simplify the following complex numbers and find their modulus. If the real part of z+1/z+I is 1, then find the locus of z​

Answer 1

Answer :-

Locus of Z is a straight line.

Step-by-step explanation :-

Given :

Real part of $\bold{\frac{z+1}{z+i}}$ is 1.

To find :

Locus of z.

Solution :

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

$\rightarrow \bold{\frac{[(x+1)+iy][x-i(y+1)}{x^2+(y+1)^2}}$

$\rightarrow \bold{\frac{x[(x+1)+y(y+1)}{x^2+(y+1)^2}}+\bold{\frac{i(x+1)(y+1)+xy)}{x^2+(y+1)^2}}$

Given real part is 1.

$\rightarrow \bold{\frac{x^2+x+y^2+1}{x^2+y^2+2y+1}}=\bold{1}$

$\rightarrow \bold{x^2+x+y^2+y}=\bold{x^2+y^2+2y+1}=\bold{x-y=1}$

∴ Locus of Z is a straight line.